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wrrts\jvT\j 


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short  ways  of  computing  emplbyed  by  the 
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A  Text-Book  on  Commercial  Law. 

A  Manual  of  the  Fundamental  Principles  Governing  Business 
Transactions.  For  the  Use  of  Commercial  Colleges,  High 
Schools  arid  Academies.  By  Saxter  S.  Clark,  Counsellor-at- 
Law.  Reviser  of  Young's  Government  Clasa-Book.  Handsomely 
printed.     12mo.     300  pp. 

The  design  of  the  author  In  this  volume  has  been  to  present  Pimply, 
and  compactly,  the  principles  of  law  aflFectingr  the  ordinary  transactions  of 
commercial  life,  In  the  form  of  a  Class-book  for  Schools  and  Commercial 
Colleges. 
The  plan  of  the  book  is  as  follows  : 

After  a  short  introduction  upon  the  relations  of  National  and  State  law, 
and  of  constitutional,  statute,  and  common  law,  it  is  divided  into  two  parts. 
Part  I.  treats  of  principles  applicable  to  all  kinds  of  business,  in  tiiree  divi- 
sions, treating  respectively  of  Contracts,  Agency  and  Partnership,  with  a 
fourth  division  embracing^the  subject  of  Corporations,  and  a  few  others 
general  in  their  nature.  Part  II.  takes  up  in  order  the  most  prominent 
kinds  of  business  transactions,  paying  chief  attention  to  the  subjects,  Sale 
of  Goods,  and  Commercial  Paper,  and  Is  to  a  large  extent  an  application  of 
the  principles  contained  in  the  preceding  part. 

The  chief  aim  has  been  throughout  to  make  it  a  book  practically  use- 
ful, and  one  easily  taught,  understood  and  remembered.  As  subserving 
those  purposes  attention  may  be  called  to  the  following  features  among 
others-— the  use  of  schemes  in  graded  type,  which  summarizing  a  suliject 
Impresses  it  upon  the  mind  through  the  eye;  the  summaries  of  leading 
rules  at  different  points :  a  table  of  definitions :  the  forms  of  business 
papers  most  frequently  met  with ;  and  the  frequent  use  of  examples  and 
cross-references. 

The  work  is  used  in  nearly  all  of  the  leading  Commercial  Colleges  of  the 
country.  

RECOMMENDATIONS. 

FmrnJi.  F.  Moore,  A.M.,  Pre/t.  Southern  Business  University.  Atlanta,  Oa. 
I  find  the  work  fully  adapted  for  use  in  buRiness  schools  an  a  text  book,  on 
account  of  its  conciseness  ;  also  to  the  accountant  as  a  book  of  reference  on  points 
of  commercial  law  and  business  forms.    It  is  the  most  complete  and  concise  work 
on  the  subject  that  I  have  peen. 

Soubeb's  Chicago  Business  Collkgr,  Chicago,  111.,  Aug,  14, 1883. 
Send  to  my  address,  by  freight,  200  Clark's  Commercial  Law. 

J.  J.  SOUDER,  Prop'r. 

Spencerian  Business  College,  Milwaukee,  Wis.,  Aug,  1, 1882, ' 
Please  forward  me,  by  express,  100  copies  Clark's  Commercial  Law. 

II.  C.  SPENCER,  Principal. 
The  B.  and  S.  Davenpobt  Business  College,  Davenport,  Iowa,  Nov.  25,  1882. 
You  may  ship  us,  by  freight,  120  Clark's  Commercial  Law. 

LILLIBRIDGE  &  VALENTINE,  Principals. 

Metropolitan  Business  College,  Chicago,  111.,  Aug.  8,  1882. 
Please  ship  us  150  Clark's  Commercial  Law,  HOWE  &  POWERS,  Prop'rs, 

Lawrence  Business  College,  Lawrence,  Kan.,  Aug,  25, 1882. 
Please  send  us  100  copies  of  Clark's  Commercial  Law. 

BOOR  &  McILRAVY,  Prop'rs. 
New  Jersey  Business  College,  Newark,  N.  J.,  Sept.  22,  1882. 
Please  send  us,  by  express,  60  Clark's  Commercial  Law. 

MILLER  &  DRAKE,  Priricipals. 

EFflNGHAM  MAYNARD  &  CO..  Publishers,  New  York. 


w^:^<m^^^m 


j(y^g^.y.^,^-t/Tr&-^^^^  ''=^^  — 


THOxMSON'S    NEW    SERIES    OF    MATHEMATICS. 


NEW 

PRACTICAL  ALGEBRA; 

ADAPTED     TO 

THE  IMPROVED  METHODS  OF  INSTRUCTION 

IN 

SCHOOLS,   ACADEMIES,   AND    COLLEGES 
WITH   AN   APPENDIX. 

BT 

JAMES    B.    THOMSON,    LL.D., 

AUTHOR   OF   A   SERIES    OF   MATHEMATICS. 


Lb=SSSi^      NEW   YORK: 

Effingham    Maynard    &    Co., 

SUCCESSORS  TO 

Clark  &  Maynakd,  Publishers, 

771  Broadway  and  67  &  69  Ninth  St. 

1889. 


THOMSON'S  Mathematical  Series. 


I.  A  Graded  Series  of  Arithmetics,  in  three  Books,  viz. : 

Naw  Illustrated  Table  Book,   or  Juvenile  Arithmetic.      With  oral 
and  slate  exercises.     (For  beginners.)     128  pp. 

New   Rudiments    of  Arithmetic.      Combining  Mental  with  Written 
Arithmetic.     (For  Intermediate  Classes.)    224  pp. 

New  Practical  Arithmetic.    Adapted  to  a  complete  business  education. 
(For  Grammar  Departments.)    384  pp. 

n.  Independent  Books, 

Key  to  New  Practical  Arithmetic.     Containing  many  valuable  sug- 
gestions.    (For  teachers  only.)    168  pp. 

New^  Mental   Arithmetic.      Containing    the   Simple   and    Compound 
Tables.     (For  Primary  Schools.)     144  pp. 

Complete  Intellectual  Arithmetic.      Specially  adapted  to  Classes  in 
Grammar  Schools  and  Academies.     168  pp. 


in.  Supplementary  Course. 

New  Practical  Algebra.     Adapted  to  High   Schools  and  Academies. 
312  pp. 

Key  to  New  Practical  Algebra.    With  full  solutions.     (For  teachers 
only.)    224  pp. 

New  Collegiate   Algebra.     Adapted  to  Colleges  and  Universities.   By 
Thomson  &  Quimby.    346  pp. 

Complete  Higher  Arithmetic.     (In  preparation.) 

\*  Each  book  of  the  Series  is  complete  in  itself. 


Copyright,  1879,  1880,  by  James  B.  Thomson. 
^lectrotyped  by  Smith  8c  McDougal,  82  Beekman  St.,  New  Yo'k. 


PREFACE. 


XT  has  loDg  been  a  favorite  plan  of  the  author  to  make  a 
-*-  Practical  Algebra — a  Book  combining  the  important 
principles  of  the  Science,  with  their  application  to  methods 
of  business. 

Several  years  have  elapsed  since  he  began  to  gather  and 
arrange  materials  for  this  object.  Many  of  the  more 
important  parts  have  been  written  and  re-written  and 
again  revised,  till  they  h^ve  found  embodiment  in  the  book 
now  offered  to  the  public. 

In  the  execution  of  this  plan,  clearness  and  brevity  in  the 
definitions  and  rules  have  been  the  constant  aim. 

A  series  of  practical  problems,  applying  the  principles 
already  explained,  has  been  introduced  into  the  fundamental 
rales,  thus  relieving  the  monotony  of  the  abstract  operations, 
and  illustrating  their  use. 

The  principles  are  gradually  developed,  and  explained  in 
a  manner  calculated  to  lead  the  pupil  to  a  full  understanding 
of  the  difficulties  of  the  science,  before  he  is  aware  of  their 
existence. 

The  rules  are  deduced  from  a  careful  analysis  of  practical 
problems  involving  the  principles  in  question — a  feature  so 
extensively  approved  in  the  author's  Series  of  Arithmetics. 

184012 


iv  PREFACE. 

The  arrangement  of  subjects  is  consecutive  and  logical, 
their  relation  and  mutual  dependence  being  pointed  out  by 
frequent  references. 

The  examples  are  numerous,  and  have  been  selected  with 
a  view  to  illustrate  and  familiarize  the  principles  of  the 
Science;  while  puzzles,  calculated  to  waste  the  time  and 
energy  of  the  pupil,  have  been  excluded. 

Special  attention  has  also  been  given  to  Factoring, 
Generalization,  and  the  appHcation  of  Algebra  to  business 
Formulas. 

In  these  and  other  respects,  it  is  believed,  some  advance 
is  made  beyond  other  books  of  the  kind.  While  adapted 
to  beginners,  it  covers  as  much  ground  as  the  majority  of 
students  master  in  their  Mathematical  course. 

In  presenting  this  book  to  the  public,  the  author  ventures 
to  hope  it  may  receive  the  approval  so  generously  bestowed 
upon  his  former  publications. 

J.  B.  THOMSON. 

Brookltn,  N.  Y.,  Sept,  1877. 


NOTE. 

At  the  suggestion  of  several  teachers,  an  Appendix  has  been  added 
to  the  present  edition  of  the  Practical  Algebra,  containing  a  selection 
of  College  Examination  Problems  used  for  admission  to  Yale,  Harvard, 
and  other  colleges.  These  are  preceded  by  a  collection  of  examples  of 
a  similar  character,  calculated  to  make  experts  in  Algebra. 


CONTENTS. 


Introduction^  ....•..-9 
Definitions,     ---••-•-  9 

Algebraic  Notation,    ----••-9 

Algebraic  Operations,      -----.         14 

Classification  of  Algebraic  Quantities,         -        -        •    ^9 
Force  of  the  Signs,  -        -        -        -        -        -        -        21 

Axioms,      --.---•--22 

Addition,      -.- 23 

Subtraction, 29 

Applications  of  the  Parenthesis,      •       "        "        '        SS 

Multiplication,      -------35 

Demonstration  of  the  Rule  for  Signs,  "  -  "  37 
Multiplying  Powers  of  the  Same  Letter,  -  •  -  38 
Principles  and  Formulas  in  Multiplication,  •  -  42 
Problems,  ---------44 

Division, -       46 

Cancelling  a  Factor,  -  --  •  -  •  -46 
Signs  of  the  Quotient,  -  -  •  -  -  -  47 
Dividing  Powers  of  the  Same  Letter,  -  -  -  -  48 
Dividing  Polynomials,  -  -  -  -  -  •  50 
Problems,   -        -        -        -        -        -        -        -        -51 

Factoriuff,  ----..--53 
Prime  Factors  of  Monomials,  -  -  -  -  -  54 
Greatest  Common  Divisor  of  Polynomials,       ^        -        (>Z 

Demonstration, 6^ 

Least  Common  Multiple  of  Polynomials,  -        •        69 


6  CONTENTS, 


PAGE 


Fractions, •..yo 

Signs  of  Fractions,  --.--..71 
Reduction  of  Fractions,  -----  .  y^ 
Common  Denominators,  -  -  -  •  -  -  77 
Least  Common  Denominator,  -  -  •  -  -  7^ 
Addition  of  Fractions,     -        -        -        -  -        80 

Subtraction  of  Fractions,  ---.--  82 
Multiplication  of  Fractions,  -  •  •  -  -  84 
Division  of  Fractions,  --.••-    89 

Simple  Equations, 95 

Transposition,  -----.--96 
Reduction  of  Equations,  --        -        -        -        -        97 

Simiittaneons  Equations,  -  -  -  -  112 
Elimination  by  Comparison,  -  -  -  -  -113 
Elimination  by  Substitution,  -  -  -  -  -114 
Elimination  by  Addition  or  Subtraction,  -  -  -  115 
Three  or  More  Unknown  Quantities,  -        -        -        -  120 

Generalir^ation, 124 

Formation  of  Rules,    -        -        -        -        -        -        -126 

Generalizing  Problems  in  Percentage,     -        -        -      128 
Generalizing  Problems  in  Interest,      -        -        -        -  13^ 

Conjunction  of  the  Hands  of  a  Clock,    -        -        -       133 

Involution, 134 

Reciprocal  Powers, -        -135 

Negative  Exponents,  -        -        -        -        -        -        -135 

Zero  Power,     -        -        -        -        -        -        -        -136 

Formation  of  I^owers, i3<^ 

Formation  of  Binomial  Squares,      -  -        -        -       139 

Binomial  Theorem,     -        -        -        •  -        -        -140 

General  Rule, ---141 

Addition  and  Subtraction  of  Powers,  -  -        -        -  144 

Multiplication  and  Division  of  Powers,  -        -        -       145 

Changing  Sign  of  Exponent,       -        -  •        -        -  146 

Evolution, 147 

Decimal  Exponents,    --        -        -        •        -        -149 


CON^TENTS.  7 


PAGE 


Signs  of  Roots,   --------  150 

Square  Root  of  the  Square  of  a  Binomial,        -        -       151 

Square  Root  of  a  Polynomial, 152 

General  Rule, 153 

JRadical  Quantities^ 154 

Reduction  of  Radicals, 155 

Addition  of  Radicals, 159 

Subtraction  of  Radicals,  -        -        -        -        -        -160 

Multiplication  of  Radicals,  -        -----  161 

Division  of  Radicals,        -        -        -        -        -        -163 

Involution  of  Radicals,        -        -        -        -        -        -  164 

Evolution  of  Radicals,     -        -        -        --        -165 

Changing  Radicals  to  Rational  Quantities,          -        -  166 
Radical  Equations, 169 

Quadratic  Equations^ 171 

Pure  Quadratics.      -        -        -        -        -        -        -172 

Affected  Quadratics,   -        -        -        -        -        -        -175 

First  Method  of  Completing  a  Square,     -        -        -       176 
Second  Method  of  Completing  a  Square,    -        -        -  179 
Third  Method  of  Completing  a  Square,  -        -        -       180 
Problems,  --------  184 

Simultaneous  Quadratics,       -        -        -        -        -       187 

Hatio,      - 192 

Proiwrtion,       --..---      196 
Theorems,         -------    198-203 

Problems,      --------      204 

Arithmetical  I^rof/ression,  -  -  -  -  205 
The  Last  Term  of  an  Arithmetical  Series,  -  -  207 
The  Sum  of  an  Arithmetical  Series,  -  -  -  -  208 
Miscellaneous  Formulas  in  Arith.  Progression,  -  211 
Inserting  Arithmetical  Means,  -  -  -  -  -212 
Problems,      -         -         -        -        -         -         -         -212 

Geometrical  JProffression^  -  -  -  -  215 
The  Last  Term  of  a  Geometrical  Series,  -         -       216 

The  Sum  of  a  Geometrical  Series,      -        -         -         -  2 1 7 


8  CONTENTS 


PAGE 


Miscellaneous  Formulas  in  Geometrical  Progression,  -  220 
Inserting  Geometrical  Means,         -        -        -        -      221 

Problems,  -------.  221 

Harmonical  Progression,        -        -        -        -        -      223 

Infinite  Series,  -        -        -        -        -        .        .        -226 

})Ogarithnis, 232 

Finding  the  Logarithm  of  a  Number,  -        -        -  235 

To  find  the  Number  belonging  to  a  Logarithm,  -  -  236 
Multiplication  by  Logarithms,  -  -  -  -  -  237 
Division  by  Logarithms,  -  -  -  .  -  238 
Involution  by  Logarithms,  -        -        -        .        .  238 

Evolution  by  Logarithms,  -  -  -  -  -239 
Compound  Interest  by  Logarithms,  -  -  -  -  240 
Table  of  Logarithms,     -        -        -.-        -        -241 

Mathematical  Induction^        -       -       -       -  243 

Business  Formulas, 245 

Formulas  for  Profit  and  Loss,   -        -        -        -        -  245 

Formulas  for  Simple  Interest,         -        -        -        -       247. 

Formulas  for  Compound  Interest,      -         -        -         -  248 

Formulas  for  Discount,  -         -         -        -        -         -250 

Formulas  for  Compound  Discount,     -        -        -        -  251 

Formulas  for  Commercial  Discount,        -        -        -       252 
Formulas  for  Investments,         -        -         -        -        _  253 

Formulas  for  Sinking  Funds,  -        -        -        -       255 

Formulas  for  Annuities,     -        -        -        -        -        -257 

Discussion  of  ProhlemSy       -       -       -       -      261 

Problem  of  the  Couriers,  -        -        -        -        -        -262 

Imaginary  Quantities,      -        •        •        •        -         -265 
Iiidsterminate  and  Impossible  Problems,    -        •        -  267 
Negative  Solutions,  ._.---      268 

Horner's  Method  of  Approximation,  -        -        -        -  269 

Test  Examples  for  Review,      -        -        -        -        -      274 

Appendix,  ---------  283 

Collegiate  Examination  Problems,  -        -        -        -      291 

Answers,    ---------  295 


ALGEBRA 


CHAPTEE   I. 

INTRODUCTION. 

Art.  1.  Algebra*  is  the  art  of  computing  by  letters  and 
signs.    These  letters  and  signs  are  called  Symbols, 

2.  Quantity  is  anything  which  can  be  measured;  as 
distance,  weight,  time,  number,  &c. 

3.  A  Measure  of  a  quantity  is  a  unit  of  that  quantity 
established  by  law  or  custom,  as  the  Standard  Unit. 

Thus,  the  measure  of  distance  is  the  yard ;  of  weight,  the  Troy 
pound;  of  time,  the  mean  solar  day,  etc. 

NOTATION. 

4.  Quantities  in  Algebra  are  expressed  by  letters,  or 
by  a  combination  of  letters  and  figures  j  as,  a,  h,  c,  z^, 
4y,  Sz,  etc. 

The  first  letters  of  the  alphabet  are  used  to  express  known 
quantities;  the  last  letters,  those  which  are  unktiown. 

Questions.— I.  What  is  algebra  ?  Letters  and  signis  called ?  2.  Quantity?  3.  A 
measure  ?    4.  How  are  quantities  expressed  ? 

*  From  the  Arabic  al  and  gdbron,  reduction  of  parts  to  a  whole. 


10  INTEODUCTION-. 

5.  The  Letters  employed  have  no  fixed  numerical 

value  of  themselves.  Any  letter  may  represent  any  num- 
ber, and  the  same  letter  may  represent  clijfere7it  numbers, 
subject  to  one  limitation;  the  same  letter  must  always  stand 
for  the  same  number  throughout  the  same  problem. 

6.  The  delations  of  quantities,  and  the  operations  to 
be  performed,  are  expressed  by  the  same  signs  as  in  Arith- 
metic. 

7.  The  Sign  of  Addition  is  a  perpendicular  cross, 
cdXlQdi  plus ;  *  as,  +. 

Thus,  a+&  denotes  the  sum  of  a  and  &,  and  is  read,  "  a  plus  &,"  or 
••a  added  to  &." 

8.  The  Sign  of  Subtraction  is  a  short,  horizontal 
line,  called  mi?ius  j  f  as,  — . 

Thus,  a  —  b  shows  that  the  quantity  after  the  sign  is  to  be  subtract- 
ed from  the  one  before  it,  and  is  read,  " a  minus  b,"  or  "a  less  b." 

9.  The  Sign  of  Multiplication  is  an  oblique 
cross;  as,  x. 

Thus,  a  X  6  shows  that  a  and  b  are  to  be  multiplied  together,  and  is 
read,  *'a  times  &,"  "  a  into  6,"  or  "a  multiplied  by  6.'* 

10.  Multiplication  is  also  denoted  by  a  period  be- 
tween the  factors ;  as,  a  •  h. 

But  the  multiplication  of  letters  is  more  commonly  ex- 
pressed by  writing  them  together,  the  signs  being  omitted. 
Thus,  sa&c  is  equivalent  to  5  x  a  x  &  x  c. 

11.  The  Sign  of  Division  is  a  short,  horizontal 
iine  between  the  points  of  a  colon ;  as,  -^. 

Thus,  n-i-b  shows  that  the  quantity  before  the  sign  is  to  be  divided 
by  the  one  after  it,  and  is  read,  "  a  divided  by  b." 

5.  Value  of  the  letters  ?  6.  Relations  of  quantities  expressed  ?  7.  Describe  the 
sign  of  addition.    8.  Subtraction.    9.  Mntiplication.    10.  How  else  denoted  ? 

♦  The  Latin  term  plus,  signifies  more. 
\  The  Latin  minus,  signifies  it«*t 


DEFIN^ITIONS.  11 

12.  Division  is  also  denoted  by  writing  the  divisoi 
under  the  divide7id,  with  a  short  line  between  them. 

Thus,  T  shows  that  a  is  to  be  divided  by  6,  and  is  equivalent  to  a-i-h. 

13.  The  Sign  of  Equality  is  two  short,  horizontal 
lines,  equal  and  parallel;  as,  =. 

Thus,  a  =  6  shows  that  the  quantity  before  the  sign  is  equal  to  tht 
quantity  after  it,  and  is  read,  "  a  equals  h"  or  *'  a  is  equal  to  W* 

14.  The  Sign  of  Inequality  is  an  acute  angle,  with 
the  opening  turned  toward  the  greater  quantity;  as,  ><. 

Thus,  a>h  shows  that  a  is  greater  than  6,  and  a<.h  shows  that  a 
IS  less  than  h. 


15.  The    Parenthesis   (    ),   or    Vinculum  , 

indicates  that  the  included  quantities  are  taken  collectively, 
or  as  one  quantity. 

Thus,  3  (05  +  6)  and  a  +  6  x  3,  each  denote  that  the  sum  of  a  and  & 
is  multiplied  by  3. 

16.  The  Double  or  Ambiguous  Sign  is  a  combi- 
nation of  the  ^[gn^plus  and  minus;  as,  ±. 

Thus,  a±J)  shows  that  6  is  to  be  added  to  or  subtracted  from  a,  and 
is  read,  "  a  plus  or  minus  h.'* 

17.  The  character  ,*, ,  denotes  hejice,  therefore, 

18.  Every  quantity  is  supposed  to  be  preceded  by  the 
sign  plus  or  minus.  When  no  sign  is  prefixed,  the  sign  -f 
is  always  understood. 

19.  Like  Signs  are  those  which  are  all  plus,  or  oL 
minus  ;  as,    +  «  +  J  +  c,  or  —x  —  y  —  z, 

20.  Unlike  Signs  include  both  plus  and  minus;  as, 
a  —  b-\-c  and   —x  +  y  —  z, 

II.  Describe  the  sign  of  division  ?  12.  How  else  denoted  ?  13.  The  sign  of 
equality?  14.  Of  inequality?  15.  Use  of  a  parenthesis  or  vinculum  ?  16.  Double 
sign  ?  17.  Sign  for  "  hence,"  etc.  ?  18.  By  what  is  every  quantity  preceded  ?  When 
none  is  expressed,  what  is  understood  ?    10,  Like  signs  ?    20.  Unlike  * 


12  INTRODUCTION. 

21.  A  Coefficient  *  is  a  number  or  letter  prefixed  to  a 
quantity,  to  show  liow  many  times  the  quantity  is  to  be 
taken.     Hence,  a  coefiicient  is  a  multiplier  ox  factor. 

Coefficients  may  be  numeral,  literal,  or  mixed. 

Thus,  in  5«,  5  is  a  numeral  coefficient  of  a  ;  in  he,  6  is  a  literal  co- 
efficient of  c ;  in  2>dx,  ^d  is  a  mixed  coefficient  of  x. 

When  no  numeral  coefficient  is  expressed,  i  is  always 
understood. 

Thus,  a^  means  lajy. 


EXERCISES    IN     NOTATION. 

22.  To  express  a  Statement  by  Algebraic  Symbols, 

It  is  required  to  express  the  following  statement  in 
algebraic  symbols: 

1.  The  product  of  a,  I,  and  c,  divided  by  the  sum  of  c  and 
d,  is  equal  to  the  difference  of  x  and  y,  increased  by  the 
product  of  a  multiplied  by  7. 

Ans.  axbxc-T-{c  +  d)  =  {x  — y)  -{-  ja. 

Or  —77^  =  {^  —  y)  +  7«-    Hence,  the 

Rule. — For  the  words,  substitute  the  signs  which  indicate 
the  relations  of  the  quantities  and  the  oj)erations  to  be  per- 
formed. 

Express  the  following  by  algebraic  symbols : 

2.  The  sum  of  4c,  d,  and  m,  diminished  by  $x,  equals  the 
product  of  a  and  b. 

3.  The  product  of  5c  and  d,  increased  by  the  quotient  of 
a  divided  by  b,  equals  the  product  of  x  and  y. 

21.  A  (Coefficient?  When  no  coefficient  is  expressed,  what  is  understood f 
22.  How  translate  a  statement  from  common  lan^age  into  algebraic  symbols  ? 

*  Coefficient,  Latin,  con,  with,  and  efficere,  to  effect ;  literally,  a 
eo-operator. 


EXEKCISES     IK     KOTATIOK.  13 

4.  The  quotient  of  3^  divided  by  5^,  increased  by  4m, 
equals  the  sum  of  c  and  6d,  diminished  by  the  product  of 
7«  and  x. 

5.  If  to  the  difference  between  a  and  I,  we  add  the 
product  of  X  into  y,  the  sum  will  be  equal  to  m  multiplied 
by  6n. 

6.  The  difference  between  x  and  y,  added  to  the  sum  of 
\a  and  h  minus  m,  equals  the  product  of  c  and  d,  increased 
by  15  times  m. 


These  and  the  following  exercises  should  be  supplemented  by 
dictation,  until  the  learner  becomes  familiar  with  them. 

23.  To   translate   Algebraic  Expressions   Into   Common 
Language. 

Express  the  following  statement  in  common  language : 

Substituting  words  for  signs,  we  have  the  sum  of  a  and  h, 
divided  by  d,  equals  twice  the  product  of  «,  b,  and  c,  dimin- 
ished by  the  sum  of  x  and  y,  increased  by  the  quotient  of 
d  divided  by  the  product  of  a  and  h,  Ans.    Hence,  the 

EuLE. — For  the  signs  indicating  the  given  relations  and 
operations,  substitute  words. 

Express  the  following  in  common  language: 

2. \-  a  —  b  =: 1-  axy  —  ^cd, 

X  c 

Sa  ^a  —  hcd 

4o ax  -\-  be  = 7,x, 

5  X  ^       ^ 

aic  —  x^         .  cdh  +  x 


Zd      --^-   '   ^^-       2a 
xy      a—b  _x  -\-  y      2 
Sa  X  a  3c 


6    4Q^^y  ,   a—b  _x-\-  y       2a  -\-  d 


23.  How  translate  algebraic  exprespions  into  common  language  J 


14  Il^^TRODUOTION'. 


ALGEBRAIC    OPERATIONS. 

24.  An  Algebraic  Operation  is  combining  quanti- 
ties according  to  the  principles  of  algebra. 

25.  A  Theorem  is  a  statement  of  a  principle  to  be 
proved. 

25.  a.  A  Problem  is  something  proposed  to  be  done,  as 
a  question  to  be  solved. 

26.  The  Equality  between  two  quantities  id  denoted 
by  the  sign  =  .     (Art.  13.) 

27.  The  Expression  of  Equality  between  two 
quantities  is  called  an  Equation,  Thus,  15  —  3  =  7  +5 
is  an  equation. 

PROBLEMS. 

28.  The  following  problems  are  solved  by  combining  the 
preceding  principles  with  those  of  Arithmetic. 

I.  A  and  B  found  a  purse  containing  12  dollars,  and 
divided  it  in  such  a  manner  that  B's  share  was  three  times 
as  much  as  A's.     How  many  dollars  did  each  have  ? 

By  Arithmetic. — A  had  i  share  and  B  3  shares ;  now  i  share  + 
3  shares  are  4  shares,  which  are  equal  to  12  dollars.  If  4  shares  equal 
12  dollars,  i  share  is  equal  to  as  many  dollars  as  4  is  contained  times 
in  12,  which  is  3.  Therefore,  A  had  3  dollars,  and  B  had  3  times  as 
much,  or  9  dollars. 

By  ALGEBRA.~We  represent  opebatiok. 

A's   share   by  a?,  and  form  an  -t-iGt  o;  =  A  s  share, 

equation  by  treating  this  letter  then  ^X  =  B's  share, 

as  we  treat  the  answer  in  proving  and     X  -{-  2>^  ^=  12  dollars, 

an  operation.     If  x  represent  A's  ^^^^t  is,'        4:^:  =  1 2  dollars, 

share,  3a;  will  represent  B's,  and  xt  ^   i      a 

,  ^  n        .1  !>         Hence,        a;  =    3  doL,  A. 

a;+3a;=i2  dollars,   the  sum  of  '  a   ^    n 

both.      Uniting    the    terms,  we  3^  =    9  ^oL,  B. 

have  the  equation,  4a?  =  12  dollars.    To  remove  the  coefficient  4,  we 

24.  What  Is  an  algebraic  operation  ?  25.  A  problem  ?  A  solution  ?  26.  Equality 
denoted  ?    27-  Tlie  expression  of  equality  called  ? 


ALGEBRAIC     OPERATIONS.  "      15 

divide  both  sides  of  tlie  equation  by  it.  For,  if  equals  are  divided  by- 
equals,  the  quotients  are  equal.  Therefore,  :c  =  3  dollars,  A's  share, 
and  3a;  =  9  dollars,  B's  share.    (Ax.  5.) 

Proof. — By  the  first  condition,  9  dollars,  B's  share  =  3  times  3  dol- 
lars, A's  share.  By  the  second  condition,  9  doDars  +  3  dollars  = 
12  dollars,  the  sum  found.     Hence, 

29.  When  a  quantity  on  either  side  of  the  equation  has  a 
coefficient,  that  coefficient  may  be  removed,  hy  dividing 
every  term  on  both  sides  of  the  equation  hy  it. 

2.  A  and  B  together  have  15  pears,  and  A  has  twice  as 
many  as  B :  how  many  has  each  ? 

By  Algebra. — If  x  represents 
B's  number,  2.x  will  represent  A's, 
and  X+2X,  or  3a;,  will  represent 
the  number  of  both.  Dividing 
both  sides  by  the  coefficient  3,  we 
have  a;  =  5  pears,  B's  number,  and 
2X  =  10  pears,  A's. 

Note— It  is  advisable  for  the  learner  to  solve  each  of  the  follow- 
ing problems  by  Arithmetic  and  by  Algebra. 

3.  A  lad  bought  an  apple  and  an  orange  for  8  cents,  pay- 
ing 3  times  as  much  for  the  orange  as  for  the  apple.  What 
was  the  price  of  each  ? 

4.  A  farmer  sold  a  cow  and  a  ton  of  hay  for  40  dollars, 
the  cow  being  worth  4  times  as  much  as  the  hay.  What 
was  the  value  of  each  ? 

5.  The  sum  of  two  numbers  is  36,  one  of  which  is  3  times 
the  other.    What  are  the  numbers  ? 

6.  A,  B,  and  C  have  28  peaches;  B  has  twice  as  m.any  as 
C,  and  A  twice  as  many  as  B.     How  many  has  each  ? 

7.  A  father  is  3  times  the  age  of  his  son,  and  the  sum  of 
their  ages  is  48  years.     How  old  is  each  ? 

2^.  How  remove  a  coefficieijt  ? 


OFBBATION, 

Let 

X  =  B's  number; 

then 

2X  =  A's         « 

and 

3ic  =  15  pears. 

,  *, 

x=    s  pears,  B's. 

22;=  10  pears,  A's. 

16  IN^TRODUCTION". 

8.  A  and  B  trade  in  company,  and  gain  loo  dollars.  If 
A  puts  in  4  times  as  much  as  B,  what  will  be  the  gain  of 
each? 

9.  The  sum  of  three  numbers  is  90.  The  second  is  twice 
the  first,  and  the  third  as  many  as  the  first  and  second: 
what  are  the  numbers  ? 

10.  A  cow  and  calf  were  sold  for  6;^  dollars,  the  cow  being 
worth  8  times  as  much  as  the  calf.  What  was  the  value  of 
each? 

11.  A  man  being  asked  the  price  of  his  horse,  rephed 
that  his    horse,   saddle    and  bridle   together  were  worth 
126  dollars;  that  the  saddle  was  worth  twice  as  much  as 
the  bridle,  and  the  horse  7  times  as  much  as  both  the  otherr 
What  was  each  worth  ? 

12.  A  man  bequeathed  $36,000  to  his  wife,  son  and 
daughter,  giving  the  son  twice  as  much  as  the  d?jghter, 
and  the  wife  3  times  as  much  as  the  son  and  daughter. 
What  did  each  receive  ? 

13.  The  sum  of  three  numbers  is  1872*  the  second  is 
3  times  the  first,  and  the  third  equals  the  o^ner  two.  What 
are  the  numbers  ?  . 

POWERS    AND    ROOTS. 

30.  A  JPower  is  the  product  of  two  or  more  equal 
factors. 

Thus,  the  product  2  x  2,  is  the  sqttare  or  second  power  of  2  ; 
a;  X  a;  X  a;  is  the  cube  or  third  power  of  x. 

31.  The  Index  or  JEx2Jonent  of  a  power  is  a  figure 
or  letter  placed  at  the  right,  above  the  quantity. 

Thus,  a'  denotes  a,  or  the  first  power. 

a^       "      ax  a,  the  square,  or  second  power. 
a'       **      ax  ax  a,  the  cube,  or  third  power,  etc. 

32.  A  Hoot  is  one  of  the  equal  factors  of  a  quantity. 

30.  What  is  a  power?    31.  How  denoted? 


ALGEBRAIC      EXPRESSIONS.  17 

33.  Roots  are  denoted  by  the  Madical  Slf/n  ^ 

prefixed  to  the  quantity,  or  by  a  fractional  exponent  placed 
after  it. 

Thus,  -\/a,  0^,  or  ^a  denote  the  square  root  of  the  quantity  a ; 
^ a  shows  that  the  cube  root  of  a  is  to  be  extracted,  etc. 

34.  The  Tudedc  of  the  Hoot  is  the  figure  placed 
over  the  radical  sign.  The  index  of  the  square  root  is 
usually  omitted. 

(For  negative  indices,  see  Arts.  256,  258,) 

Eead  the  following  examples : 

1.  «2  +  3«.  7.  4(a  —  hf. 

2.  ¥  —  c2.  8.  «2  ^  2ah  +  52. 

3.  «  +  52  __  ^,  g^  Va  +  h, 

4.  01^  —  y  +  y^*  10.  's/a^  —  ^. 

5.  2y^  ■\-  ^  —  z,  II.  2«^  +  c^ 

6.  3(a2^^>).  12.  4x^  +  2y\ 

Write  the  following  in  algebraic  language : 

13.  The  square  of  a  plus  the  square  of  b, 

14.  The  square  of  the  sum  of  a  and  i. 

15.  The  sum  of  a  and  b,  minus  the  square  of  c. 

16.  The  square  root  of  a,  plus  the  square  root  of  x, 

17.  The  cube  root  of  x,  minus  the  fifth  power  of  y. 

18.  The  cube  root  of  a,  plus  the  square  of  i. 

ALGEBRAIC    EXPRESSIONS. 

35.  An  Algebraic  Expression  is  any  quantity  ex- 
pressed in  algebraic  language  ;  as,  30^,  5«  —  7^,  etc. 

36.  The  Terms  of  an  algebraic  expression  are  those 
parts  which  are  connected  by  the  signs  +  and  — . 

Thus,  ina+h,  there  are  two  terms  ;  in  ic+y  x  z—a  there  are  three. 

32.  A  root?  33.  How  denoted?  34.  What  is  the  fi^ire  placed  over  ft?  rac^iQ^ 
$ipi  caU'-l  ?    35-  Wbat  is  an  alget>r8i<?  expression  ?    36.  Xts  t^riug  t 


18  Il^^TRODUCTION". 

Note. — Letters  combined  by  the  signs  x  or  -?-  do  not  constitute 
separate  terms.  Such  a  combination,  to  form  a  term,  must  have  the 
sign  +  or  —  prefixed  to  it,  and  the  operations  indicated  b,v  the  signs 
X  or  -f-  must  be  performed  before  the  terms  can  be  added  to  or  sub- 
tracted fiom  the  preceding  term.  (Art.  36.)  Thus,  a  +  bxc  has  two 
terms,  hxc  foiming  one  term  and  a  the  other. 

37.  The  Dimensions  of  a  term  are  its  several  literal 
factors. 

38.  The  Degree  of  a  term  depends  on  the  number  of 
its  literal  factors,  and  is  always  equal  to  the  sum  of  theii 
exponents. 

Thus,  cib  contains  two  factors,  a  and  &,  and  is  of  the  second  degree. 
a^x  contains  three  factors,  a,  a,  and  x,  and  is  of  the  third  degree. 
¥^  contains  five  factors,  6,  b,  x,  x,  x,  and  is  of  the  fifth  degree. 

39.  The  Numerical  Value  of  an  algebraic  expres- 
sion is  the  number  which  it  represents  when  its  terms  are 
combined  as  indicated  b}-  the  signs.     (Art.  36.) 

40.  To  Find  the  Numerical  Value  of  an  algebraic  expression. 

1.  If  a  =  s,  ^  =  7,  and  x  =  g,   what  is  the  value  of 

6«  +  85  +  3^  ? 

Analysis.  —  Since  a  =  5,  6a  must 
equal  6  times  5,  or  30  ;  since  &  =  7,  8& 
must  equal  8  times  7,  or  56 ;  and  since 
x  —  g,  3X  must  equal  3  x  g,  or  27.  Now 
30+56  +  27=113.  Therefore,  the  value 
of  the  given  expression  is  113.  Hence, 
the 

EuLE. — For  the  letters,  substitute  the  figures  which  the 
letters  represent,  and  perform  the  operations  indicated  by 
the  signs. 

2.  If  5  =  3,  c=  s,  and  d=z8,  what  is  the  value  of 
Sb-{-7c  +  6d? 

Suggestion.    15  +  35  +  48  =  98,  ^tw. 

37.  The  dimensions  of  a  term?  38.  Degree?  39.  Numerical  valae  of  an  alge* 
bra ic  expression ?    40.  Hqw  found? 


OPERATION. 

6a  =  5  X  6    = 

Sb  =  7  xS    = 

30 
56 

SX  =  SX9    = 
Ans, 

27 
"3 

ALGEBKAIC     QUAI^TITIES.  19 

Find  the  numerical  value  of  the  following  expressions, 
when  a  =  2,  J  =  3,  c  =  4,  J  =  5,  and  x=z6, 

3.  4a  +  6ah  +  5c  =  how  many  ?  Ans.  64. 

4.  (a  -\-b)  X  c  xd  —  (x-^  c)  =  how  many ? 

5.  (:f  —  a)  +  rt2;4-  (c  -j-  «)  =  how  many  ? 

6.  2;  -f-  2  +  (6/  —  6')  +  Jf  —  ic  ^  how  many  ? 

7.  f^x—  («  X  c)  4-  («  X  ^)  +  a;  =  how  many? 

8.  ^  +  (:i;  X  a)  +  «  —  ii''  +  c  =  how  many? 

CLASSIFICATION    OF    ALGEBRAIC    QUANTITIES. 

41.  Quantities  in  Algebra  are  primarily  divided  into 
hnotvn  and  unknown, 

42.  A  Known  Quantity  is  one  whose  value  is  given. 
An  Unknown  Quantity  is  one  whose  value  is  not 

given. 

These  quantities  are  subdivided  into  like  and  unlike,  posi- 
tive and  negative,  simple,  compound,  monomials,  etc. 

43.  IdJce  Quantities  are  those  which  are  expressed 
by  the  same  power  of  the  same  letters;  as,  a  and  2a, 
2X^  and  x^» 

44.  Unlike  Quantities  are  those  which  are  expressed 
by  different  letters,  or  by  different  powers  of  the  same  let- 
ters ;  as  2X  and  3?/,  2X  and  x^. 

Note. — An  exception  must  be  made  in  cases  where  letters  are  re- 
garded as  coefficients.  Thus,  ax^  and  Ix'^  are  like  quantities,  when  a 
and  6  are  considered  coeflBlcients. 

45.  A  Positive  Qnantity  is  one  that  is  to  be  added, 
and  has  the  sign  +  prefixed  to  it ;  as,  40^  +  3 J- 

46.  A  Negative  Quantity  is  one  that  is  to  be  sub- 
iracted,  and  has  the  sign  —  prefixed  to  it;  as,  4a  —  3Z'. 


41.  How  are  quantities  in  Algebra  primarily  divided  ?  42.  A  known  quantity? 
Unknown?  43.  Like  quantities?  44.  Unlike?  45.  A  positive  quantity?  46.  A 
negative  ? 


20  Il^TRODUCTION". 

47.  The  terms  Positive  and  JSegative  denote  oppo- 
siteness  of  direction  in  the  use  of  the  quantities  to  which 
they  are  applied.  If  lines  running  Nortli  from  any  point 
are  positive,  those  running  South  are  negative.  If  future 
time  \& positive,  past  time  is  negative;  if  credits  ^tq positive, 
debts  are  negative,  etc. 

48.  A  Simple  Quantity  is  a  single  letter,  or  several 
letters  written  together  without  the  sign  4-  or  — ;  as,  a. 
db,  Zxy, 

49.  A  Compound  Quantity  is  two  or  more  simple 
quantities  connected  by  the  sign  +  or  -— ;  as  3«  f  4J, 
zx  —  y, 

60.  A  Monomial  *  has  but  one  term ;  as,  a,  il. 

61.  A  binomial  f  has  two  terms  ;  as,  «  -|-  J,  «  —  h. 
Notes. — i.  The  expression  a  —  6  is  often  called  a  reddual,  because 

It  denotes  that  which  remains  after  a  part  is  subtracted. 
2.  A  binomial  is  sometimes  called  a  polynomial. 

62.  A  Trinomial  %  has  three  terms ;  as,  a  +  J  —  c. 

63.  A  Polynomial  \  has  two  or  more  terms ;  as, 
a  -\- h  —  c  ■\-  X. 

64.  An  Homogeneous   Polynomial   has   all   its 

terms  of  the  same  degree. 

Thus,  2ab  +  cd  +  s^V  is  homogeneous ;  but  ^ahc  ■\-c^  +  sx  is  not. 

66.  The  Reciprocal  of  a  quantity  is  a  unit  divided  by 
that  quantity. 

Thus,  the  reciprocal  of  a  is  - ;  the  reciprocal  ofa  +  biB r  • 

47.  What  do  the  terms  positive  and  negative  denote  ?  48.  A  simple  quantity.' 
49.  A  compound?  50.  A  monomial?  51.  A  binomial?  2^ote.  The  expression 
0  —  6  called?  52.  A  trinomial?  53.  A  polynomial?  54.  When  homogeneous? 
55.  The  reciprocal  of  a  quantity  ? 

*  Greek,  monos,  single,  and  nome,  terra,  having  one  term. 
f  Latin,  bis,  two,  and  nome,  name  (a  hybrid),  two  terms. 
X  Greek,  treis,  three,  and  nome,  name,  having  three  terms. 
\  Greek,  polus^  manj^,  and  noim,  name,  having  many  terms. 


FOECE     OF     THE     SIGKS.  21 


FORCE    OF    THE    SIGNS. 

56.  JEach  term  of  an  algebraic  expression  is  preceded 
by  the  sign  +  or  — ,  expressed  or  understood.     (Art.  i8.) 

The  Force  of  each  of  these  signs  is  limited  to  the  term 
which  follows  it;  as,  7  +  5— 3  =  12  —  3  =  9;  15—6  +  8 
=  9  +  8=17. 

57.  If  a  term,  preceded  by  the  sign  +  or  — ,  is  combined 
with  other  letters  by  the  sign  x  or  -^,  each  of  these  let- 
ters forms  a  part  of  that  term,  and  the  operations  indicated, 
taken  in  their  order,  must  be  performed  before  any  part  of 
the  term  can  be  added  to  or  subtracted  from  any  other  term. 

Tims,  the  expression  12  +  4  x  2,  shows  that  4  is  to  be  multiplied  by 
2  and  the  product  added  to  12,  and  is  equal  to  20, 

In  like  manner,  the  expression  16  —  8  -f-  2,  shows  that  8  is  to  be 
divided  by  2  and  the  quotient  subtracted  from  16,  and  is  equal  to  12. 

58.  If  two  or  more  terms  joined  by  +  or  —  are  to  be 
subjected  to  the  same  operation,  they  must  be  connected  by 
a  parenthesis  or  vinculum. 

Thus,  if  a  +  &  or  a  —  6  is  to  be  multiplied  or  divided  by  c,  the  oper- 
ations are  indicated  by  {a+b)x  e,  or  c{a+b);  (»  —  &)-*-  c,  or  • 

c 

EXERCISES. 

!•  50  +  5x2=  what  number  ? 

2.  50  —  5x2  =  what  number  ? 

3.  ac  +  4b  X  2  =  what  ? 

4.  ^b  —  6d  -T-  3  =  what  ? 

5-  15  +  5  X  3  +  lo-^  2  =what? 

6.  18  —  2x4-^2  +  10  =  what  ? 

7.  3X+Sy-i-4-\-axb  =  what  ? 

8.  6b  —  jc  X  X  +  ga  -^  S  =  what  ? 

9.  {b  -\-  c)  X  xy  =  what  ? 

56.  By  what  are  all  algebraic  terms  preceded  ?  The  force  of  each  of  these  signs  ? 
57.  Of  the  figns  x  and  n-  ?    58.  Of  the  parenthesis  and  vinculuia  ? 


22  H^TTEODUCTIOS". 

la     ^x  X  S^  -^  2Z  -{-  a  =  what  ? 

11.  {b  —  a)  -T-  xy  -]-  2Z  =  what  ? 

12.  3a;  +  a;^  +  22  X  32/  =  what  ? 

13.  The  difference  of  x  and  y,  multiph'ed  by  a  less  b,  and 
the  product  divided  'bjd  =  what  ? 

Find  the  value  of  the  following  expressions,  in  which 
a  =  s>  ^  =  4}  (^  =  2,  X  =  6,  y  =  8,  and  z=  10: 

14.  a-\-(axx)-^C'\-yxz  =  what  ? 

15.  2^  ~  (a;  —  Z»)  4-  «  X  ^  X  y  +  2;^  =  what  ? 

AXIOMS. 
59.  An  Aociom,  is  a  self-evident  truth. 

1.  Things  which  are  equal  to  the  same  thing,  are  equal  to 
each  other. 

2.  If  equals  are  added  to  equals,  the  sums  are  equal. 

3.  If  equals  are  subtracted  from  equals,  the  remainders 
are  equal. 

4.  If  equals  are  multiplied  bj  equals,  the  products  are 
equal. 

5.  If  equals  are  divided  by  equals,  the  quotients  are  equal. 

6.  If  a  quantity  is  multiplied  and  divided  by  the  same 
quantity,  its  t^^^^^^e  is  not  altered. 

7.  If  the  same  quantity  is  added  to  and  subtracted  from 
another  quantity,  the  value  of  the  latter  is  not  altered. 

8.  The  whole  is  greater  than  its  part. 

9.  The  whole  is  equal  to  the  sum  oi  all  its  parts. 

10.  Like  powers  and  like  roots  of  equal  quantities,  are 
equal. 

Note. — The  importance  of  tliorougUy  understanding  the  defini- 
tions and  principles  cannot  be  too  deeply  impressed  upon  the  mind  of 
the  learner.  The  questions  at  the  foot  of  the  page  are  designed  to 
direct  his  attention  to  the  more  important  points.  Teachers,  of  course, 
will  not  be  confined  to  them. 


CHAPTER    II. 
ADDITION. 

60.  Addition  in  Algebra  is  uniting  two  or  more  quan 

fcities  and  reducing  them  to  the  simplest  form. 

61.  The  Mesult  is  called  the  Sum  or  Amount. 

62.  Quantities  expressed  by  letters  are  regarded  as 
concrete  quantities.  Hence,  their  coeflBcients  may  be  added, 
subtracted,  multiplied,  and  divided  like  concrete  numbers. 

Thus,  ja  and  4«  are  ya,  46  and  5&  are  gb,  as  truly  as  3  apples  and 
4  apples  are  7  apples,  or  as  4  bushels  and  5  bushels  are  9  bushels. 

PRIi\!CIPLES.* 

63.  1°.  Like  quantities  only  can  he  united  in  one  term. 
2°.  Tlie  sum  of  two  or  more  quantities  ii  the  same  in 

whatever  order  they  are  added. 

CASE    I. 

64.  10  Add  like  Monomials  which  have  like  signs. 

I.  What  is  the  sum  of  i$ab  +  isao  +  igab  ? 

Analysis.— These  terms  are  like  quantities  operation. 

and  have  like  signs.     (Art.  19.)    We  therefore  +  ^S^^ 

add  the  coefficients,  to  the  sum  annex  the  com-  _j-  i^ab 

mon  letters,  and  prefix  the  common  sign.    The  1    tq^A 


result,  +  4706,  is  the  answer  required.    (Ax.  g.) 


47 « J,  Ans. 


60.  What  is  addition  ?    61.  The  result  called  ?    62.  How  are  quantities  expressed 
by  letters  regarded  ?    63.  First  principle  ?    Second  ? 

*■  The  expressions  1°,  2°,  3°.  etc..  denote  first,  second,  third,  etc. 


24  ADDITION. 

2.  What  is  the  sum  of  —  142^^,  —  i6a;y,  and  —  i8a:y  ? 

Analysis. — Since  these  terms  are  like  quan-  —  14^^ 

tities,  and  have  like   signs,  we  add  them  as  i6xil 

before,  and  prefix  the  sign  —  to  the  result,  for  r> 

the  reason  that  all    the  quantities    have    the  — 

sign  — .     Hence,  the  —  A^xy,  Ans. 

KuLE. — Add  the  coefficients;  to  the  sum  annex  thecoma 
man  letters,  and  prefix  the  common  sign. 


(3.) 

(4.) 

(S) 

(6.) 

(7.) 

sab 

s«y 

7a' 

—  'jhcd 

-4^f 

sal) 

8x1/ 

3a» 

—  Zbcd 

-3^^^ 

6ai 

xy 

4^2 

—  Sbcd 

-   a^y^ 

yah 

3«y 

a^ 

—  Ucd 

-Sa^y^ 

8.  Add  5«&2  ^  i>ja¥+  i^al^. 

9.  Add  —  Sabx^y^  —  saix^y^—  2Sabx^y\ 

10.  Add  s^dm^  +  Wdm^  +  9l^dm^  4-  W^dm^ 

11.  If  3a  -f  5«  +  «  +  7a  =  48,  to  what  is  a  equal  ? 
Solution.    3a  +  sa  +  a+'ja=i6a  ;  hence,  a=4S-7-i6,  or  3.  Ans. 

12.  If  4bc  +  ghc  +  2hc  +  ^bc  =  80,  to  what  is  he  equal? 

13.  If  xy  +  s^y  +  s^y  +  4xy  =  65,  to  what  is  xy  equai  ? 

CASE    II. 
65.  To  Add  like  Monomials  which  have  Unlike  signs. 

14.  What  is  the  sum  of  ^ab  —  ^ab  —  yah  +  gab  +  tab 
^Sab? 

Analysis, — For  convenience  in  operation. 

adding,  we  write  the  negative  terma  Sab  —    S^b 

one  under  another  in    the  right-  gab  —     jab 

hand  column,  with  the  sign  —  be-  ^^j 3^J 

fore  each,  and  the  positive  terms  ; T" 

in  the  next   column  on  the  left.  ^^^^  "  ^^^^  =  ^^^^  ^^^^^ 
We  then  find  the  sum  of  the  coefficients  of  the  positive  and  negative 

64.  How  add  lT!o^omia^•  which  have  like  signs  ? 


ADDITION.  *Z5 

terms  separately  ;  and  taking  the  less  sum  from  the  greater,  the  result 
2ab,  is  the  answer.     Hence,  the 

KuLE. — I.  Write  the  positive  and  negative  terms  in  sepa- 
rate columns  with  their  proper  signs,  and  find  the  sum  of  the 
coefficiefits  of  each  column  separately, 

II.  From  the  greater  subtract  the  less  j  to  the  remainder 
prefix  the  sign  of  the  greater,  and  annex  the  common  letters. 

Note — If  two  equal  quantities  have  opposite  signs,  they  balance 
each  other,  and  may  be  omitted. 

15.  Add  4d  +  ^d  —  ^d  -\-  6d  —  2d.  Ans.  6d. 

16.  Add  —  ^x  -}-  6x  -\-  8x  —  ^x  -\-  gx  —  yx. 

17.  Add  3«Jc  +  i2aic  —  6abc  +  s^^^o  —  loaic  —  saic. 

18.  Add  2J  —  5^  +  45  —  65  —  yb. 

19.  Add  —6g-\-4y^Sy  —  gg-\-Sy  —  y. 

20.  Add  4m  -\-  i6m  —  8m  —  9m  +  5m  —  10m. 

21.  If  6ab  +  i4ab  —  yab  +  i^ab  —  i2ab  +  i6ab  =  32,  to 
what  is  ab  equal  ? 

22.  To  what  is  bed  equal,  if  bed  —  $bcd  +  4bcd  +  4bcd 

-Sbcd=7s^ 

Eemark. — The  sum  in  Arithmetic  is  always  greater  than  any  of  its 
parts.  But,  in  Algebra,  it  will  be  observed,  the  sum  of  a  positive 
and  negative  quantity  is  always  less  than  the  positive  quantity.  It  is 
thence  called  Algebraic  Sum, 

66.  Unlike  Quantities  cannot  be  united  in  one  term^ 
Their  sum  is  indicated  by  writing  them  one  after  another, 
with  their  proper  signs.     (Art.  6^,  Prin.  i.) 

Thus,  the  sum  of  yg  and  3(?  is  neither  log  nor  lod,  any  more  than 
7  guineas  and  3  dollars  are  10  guineas  or  10  dollars.  Their  sum  is 
79  +  3^-    (Art.  63,  Prin.  i.) 

67.  Poh/no^nials  are  added  by  uniting  like  quantities, 
as  in  adding  monomials. 

65.  How  add  monomials  having  unlike  Bigne.  ?  Bern.  What  is  true  of  the  Bura  in 
Arithmetic  ?    In  Algebra  ?    66.  How  add  unlike  quantities  ?    67.  Polynomials  ? 


26  ADDITION. 

23.  What  is  the  sum  of  the  polynomial  ^ah  —  35  +  46? 
—  3^5  —  5^^  -\-  ^x  —  c—  2d;  and  bg  ■\-  d  +  zab  -f  d  ? 

Al^ALYSIS. For  OPERATION. 

convenience,      we  3^^  —  3^  +  4^  —  3^  —  C 

write    tlie  quanti-  —  5«5+     5  —  2^+  4a; 

ties   so   that   like  ^ab  +    d  +  J^ 

tenns  shall   stand — — 

one  under  another,  -  2^>  +  3^/  +     x  +  bg  -  C,  Ans. 

and  uniting  those  which  are  alike,  the  result  is  —2h  +  2,d-{-x-\-l}g—c. 

68.  From  the  preceding  illustrations  and  principles  we 
deduce  the  following 

GENERAL    RULE. 

I.  Write  the  given  quantities  so  that  like  terms  shall  stand 
one  under  another, 

II.  Unite  the  terms  which  are  alihe,  and  to  the  result 
annex  the  unlike  terrns  with  their  proper  signs.    (Art.  65.) 

1.  Add  $a  —  ^a  -{-  6a  +  ya  -{-  ga  -\-  2b  —  ^d, 

2.  Add  ^mn  -}-  ^mn  —  ^mn  +  gmn  —  xy  ■\-  be. 

3.  Add  2,bc  —  "jbc  -\-  xy  —  mn  +  1 1  Jc  +  gbc. 

^  Add  $ab  —  37W?^  —  ab  -{•  ^ab  -\-  2Z  —  ^ab  +  ab. 

5.  Add  2t^y  —  xy  -\-  ab  —  "jxy  ■\-  b  -\-  Sxy  —  xy  -{-  i^xy. 

69.  Compound  Quantities  inclosed  in  a  paren- 
thesis, are  taken  collectively,  or  as  one  quantity.  Hence,  if 
the  quantities  are  alike,  their  coefficients  and  exponents  are 
treated  as  the  coefficients  and  exponents  of  like  monomials. 
(Art.  64.) 

6.  What  is  the  sum  of  3  {a-{-b)  +  5  (a  +  b)  +  7  {a-\-b)? 
Solution.     3  («  +  6)  and  5  (a  +  &)  and  yia  +  b)  are  1 5  (a  +  b).    An». 

f.  Add  13  («  4-  J)  -1-  15  (a  +  5)  -  7  («  +  h). 

8.  Add  2>c{x  —  y)  +  ic^—y)  —  So{x—tj)  -\-yc{x—y). 

9.  Add  saVxy  +  saVxy  —  jaVxy  +  SaV^^y- 

10.  Add  sVa  +  3a/«  —  sVa  +  9 V«  —  3 V^. 

11.  Add  S^x  —  y  —  3^/^  —  y  +  sVx  —  y. 

68.  The  general  rule  for  addition ?  69.  How  aid  quaniities  included  in  a  paren- 
thesif  'i 


PROBLEMS.  27 

70.  The  sum  of  unlike  quantities  haying  a  common  letter 
or  letters,  may  be  expressed  by  inclosing  the  other  letters, 
with  their  signs  and  coefficients,  in  a  parenthesis,  and  an- 
nexing or  prefixing  the  common  letter  or  letters  to  the  result. 

12.  What  is  the  sum  of  5^2;  +  ^ix  —  4cx? 

Solution.    5«ic + 3bx-4cx  =  (5«  +  3&— 4^) aj,  or  x (5a  +  3b— 4c].  Ans. 

13.  Add  ja  —  6ba  +  sda  —  ^ma. 

14.  Add  ahy  +  sy  —  2cy  —  ^tny. 

15.  Add  9m  +  abm  —  ^jcm  ■\-  ^dm, 

16.  Add  iT^ax  —  2fix  -{-ex  —  ^dx  +  mx. 

1 7.  Add  axy  +  5a;^  —  cxy, 

PROBLEMS. 

71.  Problems  requiring  equal  quantities  to  be  added  to  each 

side  of  the  equation^ 

1.  A  has  3  times  as  many  marbles  as  B,  lacking  6 ;  and 
both  together  have  58.    How  many  has  each  ? 

Analysis. — If  x  represents  operation. 

B's  number,  then  will  3a;-6  Let  X  =  B's  No. ; 

represent  A's,  and  37  +  32;— 6  ^Yien  ^X  —  6  =  A's     " 

=  58,  the  sum  of  both.     To  ,  /-  o   v.  ix. 

.  j^  ,  X  4-  'XX  —  6  =  ^S,  both, 

remove   —6,  we  add.  an  eqiml 

positive  quantity  to  each  side  a;  + 3a;  —  6  +  6  =  58  +  6 

of  the  equation.    (Axiom  2.)  4^  ^^  64 

Uniting  the  terms,  we  have  x=  16,  B's  NOo 

4X  =  64,  and  x  =  16,  B's,  and  ox 6  =  42   A's    " 

3  times  16—6,  or  42  =  A's  No, 

72.  When  a  negative  quantity  occurs  on  either  side  of  an 
equation,  that  quantity  may  be  removed  by  adding  an  equal 
positive  quantity  to  both  sides. 

Note. — In  forming  the  equation,  we  treat  x  as  we  do  the  answer 
in  proving  an  operation. 

2.  A  kite  and  a  ball  together  cost  46  cents,  and  the  kite 
cost  2  cents  less  than  twice  the  cost  of  the  ball.  What  was 
the  cost  of  each  ? 

70.  How  may  the  sum  of  unlike  quantities  which  have  a  common  letter  be  ex- 


^8  ADDITION". 

3.  In  a  basket  there  are  75  peaches  and  pears  ;  the  num- 
ber of  pears  being  double  that  of  the  peaches,  wanting  3. 
How  many  are  there  of  each  ? 

4.  The  sum  of  two  numbers  is  85,  and  the  greater  is 
5  times  the  less,  wanting  5.     What  are  the  numbers  ? 

5.  A  certain  school  contains  40  pupils,  and  there  are 
twice  as  many  girls,  lacking  5,  as  boys.  How  many  are 
there  of  each  ? 

6.  If  44a;  +  6sx  —  24  =  85,  what  is  the  value  of  x? 

7.  If  jx  —  $  -{-  2X  =  60,  what  is  the  value  ofx? 

8.  If  4?/  +  2?/  +  5?/  —  7  =  70,  what  is  the  value  of  y? 

9.  The  whole  number  of  votes  cast  for  A  and  B  at  a  cer- 
tain election  was  450  ;  A  had  20  votes  less  than  4  times  the 
number  for  B.     How  many  votes  had  each  ? 

10.  The  sum  of  two  numbers  is  177  ;  the  greater  is  3  less 
than  4  times  the  smaller.     What  are  the  numbers? 

1 1.  What  is  the  value  of  ?/,  if  41/  +  32/  +  2?/  —  1 2  =  60  ? 

12.  A  lad  bought  a  top  and  a  ball  for  32  cents ;  the  price 
of  the  ball  was  3  times  that  of  the  top,  minus  4  cents. 
What  was  the  price  of  each  ? 

13.  A  man  being  asked  the  price  of  his  saddle  and  bridle, 
replied  that  both  together  cost  40  dollars,  the  former  being 
4  times  the  price  of  the  latter,  minus  5  dollars.  What  was 
the  price  of  each  ? 

14.  A  lad  spent  a  dollar  during  a  holiday,  using  three 
times  as  much  of  it  in  the  afternoon  as  in  the  morning, 
minus  4  cents ;  how  much  did  he  spend  in  each  part  of  the 
day? 

Find  the  value  of  x  in  the  following  equations : 

15-  3a;  -f  6a;  +  4a;  -f  52;  —  8  =  154.  Ans.  9. 

16.  2X  -\-  sx  ■{-  sx  —  10  =  130. 

17.  4X  +  3.r  +  7a:  —  12  =  S6. 

18.  lox  —  4X  +  gx  —  2S  =  155. 

19.  15a;  —  7a;  —  2a;  —  60  =  300. 
fio.  18a: —  4a; -f- a;— 75  =  225. 


OHAPTEE    III. 
SUBTRACTION. 

73.  Subtraction  is  finding  the  difference  between  two 
quantities. 

The  Minuend  is  the  quantity  from  which  the  subtrac- 
tion is  made. 
The  Subtrahend  is  the  quantity  to  be  subtracted. 
The  Difference  is  the  result  found  by  subtraction. 

74.  Since  quantities  expressed  by  letters  are  regarded  as 
concrete,  the  coefficient  of  one  letter  may  be  subtracted  from 
that  of  another,  like  concrete  numbers.     (Art.  62.) 

Thus,  7a  —  3a  =  4rtj ;  8&  —  5&  =  36. 

PRINCIPLES. 

75.  I®.  Like  quantities  only  can  le  subtracted  one  from 
another. 

2°.  Tlie  sum  of  the  difference  and  subtrahend  is  equal  to 
the  minuend. 

3"^.  Subtracting  a  positive  quantity  is  equivalent  to  add- 
ing an  equal  negative  one. 

Thus,  let  it  be  required  to  subtract  +4  from  6+4. 

Taking  +4  from  6  +  4,  leaves  6. 
Adding  —4  to  6  +  4,  we  have  6  +  4—4. 
But  (Ax.  7)  6  +  4—4  is  equal  to  6. 

4°.  Subtracting  a  negative  quantity  is  the  same  as  adding 
an  equal  positive  one. 

73.  Define  subtraction.  The  Minuend.  Subtrahend.  Difference.  75.  Name  the 
Qrpt  principle,    Second.   Illijstmte  Prtn.  3  upon  the  blackhoar^.    IlJustmte  Prin.  4. 


30 


SUBTRACTION". 


Thus,  let  it  be  required  to  subtract  —4  f  jom  10—4. 
Taking  —4  from  10-4,  leaves  10. 
Adding  +4  to  10—4,  we  have  10—4  +  4. 
But  (Ax.  7)  10—4  +  4  is  equal  to  10. 

Again,  if  the  assets  of  an  estate  be  $500,  and  the  liabilities  $300, 
the  former  being  considered  positive  and  the  latter  negative,  the  net 
yalue  of  the  estate  will  be  $500— $300  =  $200.  Taking  $50  from  the 
issets  has  the  same  effect  on  the  balance  as  adding  I50  to  the  liabilities 
in  like  manner,  taking  $50  from  the  liabilities  has  the  same  effect  as 
adding  $50  to  the  assets. 

76.  To  Find  the  Difference  between  two  like  Quantities. 

This  proposition  includes  three  classes  of  examples,  as 
will  be  seen  in  the  following  illustrations: 


1.  From  25a  subtract  ija. 

Remark. — i.  In  this  example  the  signs  are 
alike,  and  the  subtrahend  is  less  than  the  min- 
uend. Subtracting  a  positive  quantity  is 
equivalent  to  adding  an  equal  negative  one. 
(Prin.  3.)  We  therefore  change  the  sign  of 
the  subtrahend,  and  then  unite  the  terms 
25a— 17«  =  Sa, 

2.  From  4a  subtract  ja. 

Remark. — 2.  In  this  example  the  signs 
are  alike,  but  the  subtrahend  is  greater  than 
the  minuend.  Changing  the  sign  of  the  sub- 
trahend, and  uniting  the  terms  as  before,  the 
subtrahend  cancels  the  minuend,  and  has 
—3a  left.    (Prin.  3.) 

3.  From  45 «5  subtract  —  2gah. 
Remark. — 3.  In  this  example  the  signs 

aie  unlike.  Subtracting  a  negative  quantity 
is  the  same  as  adding  an  equal  positive  one. 
(Prin.  4.)  Changing  the  sign  of  the  sub- 
trahend and  proceeding  as  before,  we  have 
iSab  +  290*  =  7406.    Ans. 


OPERATION. 

2$a      Minuend. 
—  lya      Subtrahend. 

Set     Difference. 

in  addition.    Thus. 


OPERATION. 

4(1     Minuend. 

—  'JCt     Subtrahend, 

—  3a     Difference. 


OPERATION. 

4$ab      Minuend. 
•^  2gab      Subtrahend. 

74«^,  Ans, 


7^.  How  And  the  difference  between  two  like  quantities  ? 


SUBTRACTION.  '31 

4.  From  ^hc  +  7^  —  S^r,  take  ^bc  -{-  2d  —  4X. 

Analysis.  —  In    subtraction    of  operation. 

polynomials,     for    convenience,    we  g^c  -}-  jd  —  ^X 

place  like  terms  under  each  other.  -  j^ 2d  -\-  dX 

Then,  changing  the  signs  of  all  the  r 

terms  in  the  subtrahend,  we  unite  ^^^  +  5^  ^f  ^^^* 

them  as  before. 

77.  From  the  preceding  illustrations  and  principles  we 
deduce  the  following 

GENERAL    RULE. 

1.  Write  the  subtrahend  under  the  minuend,  placing  like 
terms  one  under  another. 

II.  Change  the  signs  of  all  the  terms  of  the  subtrahend,  or 
suppose  them  to  be  changed,  from  +  to  — ,  or  from  —  to 
+,  and  then  proceed  as  in  addition.     (Art.  75,  Prin.  3,  4.) 

Notes. — i.  Unlike  quantities  can  be  subtracted  only  by  changing 
the  signs  of  all  the  terms  of  the  subtrahend,  and  then  writing  them 
after  the  minuend.     (Art.  66.) 

2.  As  soon  as  the  student  becomes  familiar  with  the  principles  of 
subtraction,  instead  of  actually  changing  the  signs  of  the  subtrahend, 
he  may  simply  suppose  them  to  be  changed. 


EXAMPLES. 


1.  From  43c  +  d,  take  25c  +  d. 

2.  From  49a:,  take  23:^  +  3. 

3.  From  2Sxgz,  take  i4xgz. 

4.  From  —  43ab,  take  +  igab. 

5.  From  4ab,  take  —  i^al). 

6.  From  4s^y,  take  +  i6xy. 


Ans.  1 8c. 
.  262;  —  3. 


(7.)  (8.)  (9.)  (lo.) 

From        2oaG  42aa^  370^25  —  290:^ 

Take    —  23ac  ^ax^  —  i4«^J  +  15^^ 

77.  General  rule  for  subtraction  ?    Note.  How  subtract  unlike  quantitiee. 


32  SUBTRACTION. 

(II.)  (I2.)  (13,)  (14.) 

From       3 1  a^J  I  gabx^  —  ssm^x  4 1  x^y 

Take    —    ya^b  igaba^  +  44m^  ^  i2a^y 

15.  A  is  worth  $100,  and  B  owes  I50  more  than  he  is 
worth.     What  is  the  difference  in  their  pecuniary  standing  ? 

16.  What  is  the  difference  in  temperature,  when  the  ther- 
mometer stands  15  degrees  above  zero,  and  when  at  ic 
degrees  below  ? 

17.  By  speculation,  A  gained  on  a  certain  day  I275,  and 
B  lost  $145.  What  was  the  difference  in  the  results  of  their 
operations  ? 

(18.)  (19.)  (20.) 

From        'jxy  —  Sa  8J2  -f  jam  13^^  —  jy^ 

Take         ^xy  —  2a        — 5^  —  gam        —  s^^  —  ^y^  —  ^^ 

21.  From  i^ab  -{-  d  —  x,  subtract  ^m  — ^n, 

22.  From  gcd  —  ab,  take  2m  —  37^  —  4^. 

23.  From  13m  —  15,  take  —  5m  -|-  8. 

24.  From  jx^  —  ^x  +  15,  take  —  ^x^  ■\-^x-\-  15. 

25.  From  igab  —  2c  —  7<:7,  take  ^ab  —  15c  ~  Zd, 

26.  From  a,  take  b  —  c,  and  prove  the  work. 

27.  From  II  (a  -f  b),  take  5  (<^  +  b). 

28.  From  ij  (a  —  b  ■\'  x),  take  8  («  —  J  +  a;). 

29.  Subtract  —  18  (oj  +  Z>)  from  —  13  («  -|-  b), 

30.  Subtract  21  {x^  —  y)  from  14  (a;^  —  y). 

31.  A  and  B  formed  an  equal  partnership  and  made 
$1,000.  B's  share  by  right  was  $1,000  —  1500;  but  wish- 
ing to  withdraw,  he  agreed  to  subtract  $100  from  his  share. 
What  would  A's  share  be  ? 

32.  What  is  the  difference  of  longitude  between  two 
places,  one  of  which  is  23  degrees  due  east  from  the  meridian 
of  Washington,  the  other  37  degrees  due  west? 

Remaek. — The  svhtraJiend,  in  Algebra,  is  often  greater  than  the 
minuend,  and  the  difference  between  a  positive  and  negative  quantity 
greater  than  either  of  them.  It  js  thence  called  Algebraic  Dif- 
ference, 


SUBTRACTION,  33 

78.  The  Difference  of  unlike  quantities  which  have  a 
common  letter  or  letters  may  be  indicated  by  e?idosing  all 
the  other  letters,  with  their  coefficients  and  signs,  in  a 
parenthesis,  and  annexing,  qy  prefixing  the-  common  letter 
or  letters  to  the  result. 

SS'  From  $am,  take  2bm, 

Analysis.    3am  =  3^  times  m,  and  2bm  =  26  times  m ;  therefore, 

Sam—2bm  =  {3a— 2h)  m,  or  m  (3a— 2&).    Ans. 

34.  From  2l)x^,  take  cx^  —  dx\ 

35.  From  ahy,  take  cy  ■\-  dy  —  xy, 

36.  From  ic^,  take  W  —  ca\ 

37.  From  ahx,  take  ^cx  -\-  dx  ■\-  mx. 

38.  From  ^xy,  take  alxy  —  cxy  -\-  dxy, 

39.  From  5a<?  +  J/w<?,  take  3«c  —  Jc. 

APPLICATIONS    OF    THE    PARENTHESIS. 

79.  A  parenthesis,  we  have  seen,  shows  that  the  quanti- 
ties inclosed  by  it  are  taken  collectively,  and  subjected  to 
the  operation  indicated  by  the  sign  which  precedes  it. 
(Art.  15.) 

80.  A  parenthesis  having  the  sign  +  prefixed  to  it,  may 
be  removed  from  an  expression,  if  the  signs  of  the  included 
terms  remain  unchanged. 

Thus,  a— 6+(c— (Z  +  e)  =  a— 6+c— e?+e.    Hence, 

81.  Any  number  of  terms  may  be  inclosed  in  a  parenthe- 
sis and  the  sign  +  placed  before  it,  if  the  signs  of  the 
inclosed  terms  remain  unchanged. 

Thus,  a  +  &— c  + (?  =  «  +  (&— c  +  <f),  or  <?  +  &  +  (— c  +  <f). 
Note. — This  principle  affords  a  convenient  method  of  indicating 
the  addition  of  polynomials.    (Art.  67.) 

■ ttt; • 

rS.^How  subtract  unlike  quantities  having  a  cw&mon  letter  or  letters  ?  79.  What 
is  the  object  of  a  parenthesis?    80.  How  removed  when  the  sign  +  is  prefixed  to  U. 


34  SUBTRACTIOI'T. 

82.  A  parenthesis  having  the  sign  —  prefixed  to  it,  may 
be  removed  by  changing  the  signs  of  all  the  inclosed  terms 
from  +  to  —  and  —  to  +. 

Thus,  removing  it  from  tlie  equal  expressions, 

1      ,,       .     ')-=a  —  o  +  c  —  a.     Hence, 
«  —  6  —  (c2  —  c)        j 

83.  Any  number  of  terms  may  be  inclosed  by  a  paren- 
vhesis,  and  the  sign  —  placed  before  it,  if  all  the  signs  of 
the  inclosed  terms  are  changed. 

Thus,  a—l-^c—d  =  a—(b—c+d),  or  a—'b—{—c+d)y  etc 
Note. — This  principle  enables  us  to  express  a  polynomial  in  diffei* 
ent  forms  without  changing  its  value. 

1.  How  express  a  —  x  -\-  c,  using  a  parenthesis? 

Ans.  a  —  X  •\-  c  =z  a  —  {x  —  c),  or  a  —  ( —  c  +  x). 

2.  How  express  a  —  h  —  x  —  y-\-z,  using  a  parenthesis  ? 

Ans.  a  —  h  —  {x  -\-  y  —  z),  or 
a  —  b  —  (^  -\-  X  —  z),  or 
a  —  h—{—z-\-x-\-y). 

84.  When  two  or  more  parentheses  occur  in  the  same 
expression,  they  are  removed  by  the  same  rule,  beginning 
with  the  interior  parenthesis. 

Thus,  a—\h—c-{d  +  x)-\-e\=a—(h—c—d—x+e)=a—'b+c  +  d  +  x—6. 
Note. — Quantities  may  be  included  in  more  than  one  parenthesis, 
by  observing  the  preceding  rules. 

Remo^^^  the  parentheses  from  the  following  expressions-. 

3.  ah  —  {he  —  d).  Ans,  ah  —  hc  +  d. 

4.  h  —  (c  —  d  -{-  m), 

5.  c^x  —  {— y -\- ab  ^ /^d). 

6.  2«  —  [&  +  c  —  (a;  +  «/)  —  ^. 

7.  a  —  (b  —  c)  —  (a  ~  c)  ■\-  c  —  (a  —  h). 

Jg^  The  principles  governing  the  signs  in  the  use  and  removal  of 
parentheses  should  be  made  familiar  by  practice. 

82.  How  when  the  sign  —  is  prefixed  ?  83.  How  inclose  terms  in  a  parenthesis 
With  — prefixed  to  it? 


OHAPTEE    IV. 
MULTIPLICATION. 

85.  Multiplication  is  finding  the  amount  of  a  qnaii' 
tity  taken  or  added  to  itself,  a  given  number  of  times. 

Thus,  3  times  4  are  12,  and  4  taken  3  times  (4  +  4  +  4)  =  12. 

The  Multiplicand  is  the  quantity  to  be  multiplied. 
The  3Iultiplier  is  the  quantity  by  which  we  multiply. 
The  Product  is  the  quantity  found  by  multiplication. 

86.  The  Factors  of  a  quantity  are  the  multiplier  and 
multiplicand  which  produce  it. 

PRINCIPLES. 

87.  1°.  The  multiplier  must  he  considered  an  abstract 
quantity, 

2°.  The  'product  is  of  the  same  nature  as  the  multiplicand; 
for,  repeating  a  quantity  does  not  alter  its  nature, 

3°.  T/ie  product  of  two  or  more  factors  is  the  same  in 
whatever  order  they  are  multiplied. 

CASE    I. 

88.  To  Multiply  a  Motiotnial  by  a  MonotniaZ, 

I.  What  is  the  product  of  «  multiplied  by  c? 

Ans.  a  X  c,  or  ac. 

Note. — The  product  of  two  or  more  letters,  we  have  seen,  is  ex- 
pressed by  writing  them  one  after  another,  either  with  or  without  the 
sign  of  multiplication  between  them.     (Art.  10.) 

85.  Define  multiplication.  The  multiplicand.  Multiplier,  Product.  86,  Fao 
tors.    87.  Name  Prin.  x.    Prin.  3.    Prin.  3. 


36  MULTIPLICATIOir. 

2.  If  I  ton  of  iron  costs  a  dollars,  what  will  x  tons  cost  ? 
Analysis,    x  tons  will  cost  x  times  as  much  as  i  ton  ;  and  x  times 

a  dollars  are  ax  dollars.    That  is,  a  dollars  are  taken  x  times,  and  are 
equal  to  a  +  a  +  a . . ,  . ,  and  so  on  to  a;  terms. 

3.  What  is  the  product  of  4a  by  25  ? 

Analysis. — Since  each  coefficient  and  each  letter  opbbation. 

in  the  multiplier  and  multiplicand  is  a  factor,  it  fol-  4^ 

lows  that  the  answer  must  be  the  product  of  the  2  J 

coefficients  with  all  the  letters  of  both  factors  an-  .        r~T 
nexed.    Hence,  the 

EuLE. — Multiply  the  coefficients  together,  and  prefix  the 
product  to  the  product  of  the  literal  factors. 

Multiply  the  following  quantities : 

4.  4ab  by  50?,  Ans,  2oabx, 

5.  6bc  by  7«.  9.     'jxy  by  8ah» 

6.  ^abc  by  ^xy,  10.     6ac  by  jdx. 

7.  8dm  by  xy,  11.     gbd  by  6cm, 
S,    gbcd  by  'jxyz,                  12,     jxyz  by  gadf 


SIGNS    OF    THE    PRODUCT. 

89.  The  investigation  of  the  laws  that  govern  the  signs 
of  the  product,  requires  attention  to  the  following 


PRINCIPLES. 

1®.  Repeating  a  quantity  does  not  change  its  sign, 
2°.  The  sign  of  the  multiplier  shows  whether  the  repetitions 
of  the  multiplicand  are  to  he  added,  or  subtracted, 

90.  If  the  Signs  of  the  factors  are  alihe,  the  sign  of  the 
product  will  be  positive  ;  if  unlike,  the  sign  of  the  product 
will  be  negative. 

88.  How  multiply  a  monomial  by  a  monora.al  ?  89.  Name  Principle  i.  Prin.  2, 
90.  If  signs  of  factorH  are  alike,  what  ia  the  sign  ol  the  product?    If  unlike  1 


MULTIPLIC  ATIOK.  37 

91.  Demoksteation. — There  are  four  points  to  be 
proved: 

First,  That  -f-  into  +  produces  +. 

I/et  +a  be  the  multiplicand  and  +4  the  multiplier.  It  is  plain 
that  +05  taken  +4  times  is  +4*.  (Prin.  i.)  The  sign  of  the  multi- 
plier being  -r,  shows  that  the  product  +4^,  is  to  be  added,  which  is 
donf»  \}j  setting  it  down  without  changing  its  sign.    (Art.  66.) 

Second.  That  —  into  -f  produces  — . 

Let  —a  be  multiplied  by  +4.  Now  —a  taken  4  times  is  —4a ;  for 
a  negative  quantity  repeated  is  stiU  negative.  (Prin.  i.)  But  the 
sign  before  the  multiplier  being  + ,  shows  that  the  negative  product 
—  4a,  Is  to  be  added.  This  also  is  done  by  setting  it  down  without 
changing  its  sign.    (Art.  66.) 

Third.  That  4-  into  —  produces  — . 

Let  +a  be  multiplied  by  —4.  We  have  seen  above  that  +a  taken 
4  times  is  +^a.  But  here  the  sign  of  the  multiplier  being  — ,  shows 
that  the  product  +4a>  is  to  be  subtracted.  This  is  done  by  changing 
its  sign  from  +  to  — ,  on  setting  it  down.  Thus,  +a  x  —4  =  —4a. 
(Art.  77.) 

Fourth.  That  —  into  —  produces  +. 

Let  —a  be  multiplied  by  —4.  It  has  also  been  shown  that  —a 
taken  4  times  is  —4a.  But  the  sign  of  the  multiplier  being  — ,  shows 
tnat  this  negative  product  —4a,  is  to  be  subtracted.  This  is  also  done 
by  changing  its  sign  from  —  to  + ,  when  we  set  it  down.  Thus, 
—a  X  — 4  =  +4^.    (Art.  77.)    Hence,  universally, 

92.  Factors  having  like  signs  produce  -f,  and  unlike 
signs  — . 

13.  Multiply  -I-  4ab  by  ~  'jcd,  Ans.  —  2d>ahcd. 

14.  Multiply  —  <^xy  by  +  ^ad. 

15.  Multiply  +  ()db  by  +  ^dc. 

16.  Multiply  —  ^xy  by  —  i()alc. 

17.  Multiply  +  i2>aic  by  —  232:^. 

18.  Multiply  —  35^?/  by  —  272*^^. 

91.  Prove  the  first  point  from  the  blackboard.  The  second.  Third. '  Fourth 
92.  Rule  for  signs. 


38  MIJLTIPLIC  ATIOH. 

93.  When  a  letter  is  multiplied  into  itself,  or  taken  twice 
as  a  factor,  the  product  is  represented  hj  a  x  a,  or  aa\ 
when  taken  three  times,  by  aaa,  and  so  on,  forming  a  series 
of  powers.  But  powers,  we  have  seen,  are  expressed  by 
writing  the  letter  once  only,  with  the  index  above  it,  at  the 
right  hand.     (Art.  31.) 

19.  What  is  the  product  of  aaa  into  ««? 

Analysis,  aaa  y.axi=:  aaaaa^  or  a^,  Atib.  Now  aaa  =  a^,  and 
aa  =  a^ ;  but  adding  the  exponents  of  a^  and  a^  we  have  a^,  the  same 
as  before.    Hence, 

94.  To  multiply  powers  of  the  same  letter  together,  add 
their  exponents. 

Notes. — i.  All  powers  of  i  are  i. 

2.  When  a  letter  has  no  exponent,  i  is  always  understood. 

Multiply  the  following  quantities : 

20.  aV^(?  by  a^c,  Ans.  a^^d^, 

21.  2>(^Wxhj  2al^y,  Ans.  6a^¥xy. 

22.  32;?/2  by  5ic2,  ^^.     ab'^hjab^. 

23.  6a^  by  4«5^.  26.     $xyz  by  2xy, 

24.  a'^x'^y  by  a^a^y.  27.     6a^^c  by  :^a^dl^c. 

28.  If  a  =  3,  what  is  the  difference  between  sa  and  a^? 

29.  If  a;  =  4,  what  is  the  difference  between  4X  and  a^  ? 

95.  The  preceding  principles  illustrating  monomials  may 
be  summed  up  in  the  following 

EuLE. — Multiply  the  coefficients  and  letters  of  hoth  factors 
together;  to  the  product  prhj.^  . :.?  proper  sign,  and  give  to 
each  letter  its  proper  index. 

Note. — It  is  immaterial  in  what  or  2  the  factors  are  taken,  but  it 
is  more  convenient,  and  therefore  cusix>mary,  to  arrange  the  letters  in 
alphabetical  order.    (Art.  87,  Prin.  3.) 

30.  Multiply  —  z'^y  ^y  —  2a;. 

31.  Multiply  6^2^  by  —  ^c^lc. 

94.  How  multiply  powers  of  the  same  letter  together?  05.  What  is  the  rule  for 
multiplying  moaomifids  ? 


MULTIPLICATIOK.  39 

(32.)  {33)  (34.)  (35) 


Multiply        4x7/ 
By                 x^y 

7«^ 

5^y 

(36.) 
Multiply        s^y 
By           -  -xf 

(37.) 

7a5c3 

CASE    II. 

(38.) 

—  7ac 

(39.) 
xyz 

96.  To  Multiply  a  Polynomial  by  a  Monomial, 

1.  What  is  the  product  of  a  +  5  multiplied  by  J  ? 
Analysis.— Multiplying  each  term  of  the  operation. 

multiplicand  by  the  multiplier,  we  have  axb  Ct  -\-  0 

=  fl*,  and  &  X  &  =  &2.    The  result,  «d + &^  is  the  ^ 

product  required.    Hence,  the  Ans,  ah  A-  V^ 

Rule. — Multiply  each  term  of  the  multiplicand  ly  the 
multiplier;  giving  each  jtartial  product  its  proper  sign,  and 
each  letter  its  proper  index. 

Multiply  the  following  quantities : 

2.  he  —  adhj  ah,  Ans,  al^c  —  a%d, 

3.  3«a^  -f  4cd  by  2c. 

4.  5«^  —  2cd  -\-  xhj  lax, 

5.  4«2  —  3^5  _|-  ^2  ]3y  __  2.hd. 

6.  3^2  —  4W  —  2C^  by  —  $a^c, 

CASE    III. 

97.  To  Multiply  a  Polynomial  by  a  Polynomial, 

7.  What  is  the  product  oi  a  +  h  into  «  -f-  Z*  ? 
Analysis. — Since  the  multiplicand  is  to  operation. 

be  taken  as  many  times  as  there  are  units  in  a  -j-  h 

the  multiplier,  the  product  must  be  equal  to  a  -{-  h 

a  times  a+b  added  to  6timesa  +  &.  *Now  ~2~T     k 

a  times   a+b  =  a^  +  ab,   and  b  time^  a+b  ^ 

=  +ab+b^.    Hence,  a+b  times  g;+5  must  -{-  ah  -^  i^ 

be  equal  to  a''  +  2ab+¥.  Ans.  a^  +  2^5. -f  W- 

96.  How  multip'.y  a  r;o'»\TioTninl  l)v  1  monomi:;. 


40  MULTIPLICATIO:^. 

8.  Multiply  J  4-  2e?  —  3c  by  05  4-  5. 

Analysis.  —  We  multi-  operation. 

ply  each  term  in  the  multi-  2a  -jr  i  —  $0 

plicand  by  each  term  in  the  a  -{-  b 

multiplier,   giving  to  each  2^2+    ab -^  sac 


product    the    proper    sign. 
(Art.  89.)    Finally,  we  add 


+  2ab +  ^  —  3&g 

the  partial  products,  and  the      Ans,  2a^  +  ^ab  —  ^ac  -{-  b^  —  2,bc 
result  is  the  answer  required. 


98.  The  various  principles  developed  in  the  preceding 
cases,  may  be  summed  up  in  one 


GENERAL     RULE. 

Multiply  each  term  of  the  multiplicand  by  each  term  of 
the  multiplier,  giving  each  product  its  proper  sigti,  and  each 
letter  its  proper  exponent. 

The  sum  of  the  partial  products  will  be  the  true  ^jroduct. 

Note. — For  convenience  in  adding  the  partial  products,  like  terms 
should  be  placed  under  each  other. 

Multiply  the  following  quantities: 

1.  2a-\- bhj  s^ +  y'  5«  sa  +  4b  —  chjx—y. 

2.  3^  +  4y^J  a  —  b.  6.  ^x  +  sy  +^^ya  +  b. 

3.  /^b  —  c  by  2>d  —  a,  7.  jcdx  —  ^ab  by  2m  —  $n. 

4.  6xy  —  2a  hj  b  -{-  c.  8.  Sabc  +  4m  by  $x  —  4?/. 
9.  Multiply  ^ab""  by  Sa^.  Ans.  24a^"+\ 

10.  Multiply  —  'jacff"  by  —  Sa^af.  Ans.  56a^a;"*+". 

11.  Multiply  3a Jc"  by  xyz"^. 

12.  Multiply  acd"^  by  iibcd\ 

13.  Multiply  —  ax^  by  —  ax\ 

14.  Multiply  x{a-\-  by  hy  c{a  +  by.     (Art.  15.) 

15.  Multiply  c{a^  bf  by  5  (a  —  bf. 

16.  Multiply  a{x-\-  y)""  by  be  {x  +  y)". 

17.  Multiply  3X  {a  +  by  by  —  («  +  Z>). 


98.  fiow  multiply  a  polynomial  by  a  polynomial? 


MULTIPLICATIOiq'.  41 

99.  When  the  polynomials  contain  different  powers  of  the 
same  letter,  the  terms  should  be  arranged  so  that  the  first 
term  shall  contain  the  highest  power  of  that  letter,  the 
second  term  the  next  highest  power,  and  so  on  to  the  last 
term.    This  letter  is  called  the  leading  letter, 

(i8.)  (19.) 

a  ■)-  b 4a^—$ab 

a^^2a^+  ab^  i2a^b^-\-4a'^b^ 

a^^^a^  +  sal^-^l^,  Ans.  i2aW—sa^b^—zaW,  Am. 

20.  Multiply  a^  —  ab  -{- b^  hj  a  -^  b. 

21.  Multiply  a?  —  ab  +  W  by  a^ -^  ab -^  ^. 

22.  Multiply  x^  ■\-  X  ■\-  I  by  0^  —  x  -\-  i. 

23.  Multiply  ^x^  —  2xy  +  5  by  a;^  +  2xy  —  6. 

24.  Multiply  ^ax  —  2ay  by  ^ax  +  30!^. 

25.  Multiply  d  -{- bx  \)j  d  ■{-  ex. 

100.  The  Multiplication  of  polynomials  may  be  indi- 
cated by  inclosing  each  factor  in  a  parenthesis,  and  writing 
one  after  the  other. 

Thus,  [a  +  6)  (a  +  6)  is  equivalent  to  {a  +  h)x{a  +  h). 

Note. — Algebraic  Expressiotis  are  said  to  be  developed  or 
expanded,  wlien  the  operations  indicated  by  their  signs  and  exponents 
are  performed.  , 

26.  Develop  the  expression  (a  -\- b)  {c  -\-  d). 

Ans.  ac  •\- be  ■\-  ad  -\-  bd, 

27.  Develop  {x  +  y)  {x  —  y). 

28.  Develop  {a^  +  i)  («  +  i). 

29.  Expand  {x^  +  2xy  +  y'^)  {x  +  y). 

30.  Expand  (a"^  +  b"")  (a"^  +  5"). 

31.  Expand  [x  -{-  y  -\-  z)  {x  -^  y  -{-  z). 

99.  How  arrange  different  powers  of  the  same  letter  ?  100.  How  indicate  the  mul- 
tiplication of  polynomials  ?  '' 


42  KTJLTIPLICATION". 


THEOREMS    AND    FORMULAS. 

101.  Theorem  i.—The  Square  of  the  Sum  of  two  quan- 
tities is  equal  to  the  square  of  the  first,  plus  twice  their 
product,  plus  the  square  of  the  second, 

1.  Let  it  be  required  to  multiply 
a-\-  b  into  itself. 

Analysis. — Each  term  of  the  multipli- 
cand being  multiplied  by  each  term  of  the 
multiplier,  we  have  a  times  a +  6  and  b  times 
a  +  b,  the  sum  of  which  is  a^  +  2ab  +  &*. 
Hence,  the  A71S. 

Formula.        (a  +  W  =  a^ -\-  2ab  +  l^. 

102.  Theorem  2.— The  Square  of  the  Difference  of  two 
quantities  is  equal  to  the  square  of  the  first,  minus  tivice 
their  product,  plus  the  square  of  the  second, 

2.  Let  it  be  required  to  multiply 

a  —  hhya  —  b.  a  —  b 

Analysts. — Reasoning  as  before,  the  re-  ^ 


a-Jf-b 
a-\-b 

a^-\-    ab 
+    ab+1^ 

a^  +  2a6  +  ^>2 

suit  is  the  same,  except  the  sign  of  the  mid-  ^    —    ^^ 

die  term  2ab,  which  has  the  sign  —  instead  —    ab  ■\-  1^ 

of  -I-.    Hence,  the  A7IS.    a^  —  2ab  +  ^ 

Formula.        (a  —  bf  =  a?  —  2ah-\-  b\ 

103.  Theorem  ^.—The  Product  of  the  Sum  and  Differ- 
ence of  two  quantities  is  equal  to  the  difference  of  their 
squares, 

3.  Let  it  be  required  tp  multiply 
a  -\-bhj  a  —  b.  a  +  h 

Analysis. — This  operation  is  similar  to  ^       " 


the  last  two ;  but  the  terms  -f-  ab  and  —ab,  <^  +     ^^ 

In  the  partial  products,  being  equal,  balance  ^b  fr 

each  other.    Hence,  the  Ans.     cf  —  W' 

Formula.        {a  +  h)  (a  —  h)  =  a?  —  l^. 

loi.  What  is  Theorem  i  ?    102.  Thoorom  2  ?    103.  Theorem  3  ? 


MULTIP-LI  CATION". 


43 


104.  TJie  product  of  the  sum  of  hvo  quantities  into  a  third, 
is  equal  to  the  sum  f^f  their  products. 

4.  Let  X  and  y  be  two  quantities,  whose  sum  is  to  be 
multiplied  ^y  a.    Thus, 

Tl\e  product  of  the  sum  (a; + y)  x  a  —ax-\-ay 

The  sum  of  the  products ofa;xa+yxa=:aa;+ay 
And  aic  +  ay  =  aaj + ay.    Hence,  the 

Formula.       a  (a?  +  2/)  =  aoc  +  ay. 

105.  The  product  of  the  difference  of  two  quantities  into  a 
third,  is  equal  to  the  difference  of  their  products. 

5.  Let  X  and  y  be  two  quantities,  whose  difference  is  to  be 
multiplied  by  a.     Thus, 

The  product  of  the  difference  (a;— y)  x  a  =  ax— ay 

The  difference  of  the  products  of  a;  x  a—y  x  a  =  ax— ay 
And  ax— ay  —  ax— ay.    Hence,  the 

Formula.        a(x  —  y)  =  ax  —  ay. 

Remark. — The  application  of  the  preceding  principles  is  so  frequent 
in  algebraic  processes,  that  it  is  important  for  the  learner  to  make 
them  very  familiar. 

Develop  the  following  expressions  by  the  preceding  for- 
mulas: 

{^x  —  i)  (4.r  —  1). 
(55+i)(5J+i). 
(i  _  x)  (i  -  x). 
(i  +2:z;)(i  +  2x). 
{Sb  —  3a)  {Sb  -  3a). 
{ah  +  cd)  {ah  -f  cd). 
(3«  —  2y)  (3«  +  2y). 
{x^^y){x^-y). 
{x  -  y^)  {x  —  y^). 

{20^  +  X)  (2«2  _  ^, 


I. 

(a+  i)(«+  i). 

II. 

2. 

(26?  4-   I)  (265+   l). 

12. 

3. 

i2a  —  h)  {2a  —  h). 

13. 

4. 

{x  +  y)  {x  +  y). 

14. 

5- 

(^  -y){x-  y). 

15- 

6. 

{i  +  x){i-x). 

16. 

7. 

W-y){iy'-y)' 

17. 

8. 

(4m  —  3n)  {4m  +  3^)- 

18. 

9- 

(^2_^)(^2  +  ^). 

19. 

10. 

(i  _  7:^,)  (i  _|_  7:^;). 

20. 

104.  What  is  the  product  of  the  sum  of  two  quantities  into 
105.  Of  the  difference? 


third  equal  to? 


Let    x=. 

:  No.  apples ; 

X 

:  pears. 

'<- 

24 

^x  +  x  = 

:72 

4X  = 

,',      X  = 

:72 

:  18  apples. 

X  _ 

:    6  pears. 

44  MULTIPLIC  ATIONo 


PROBLEMS. 

106.  Problems   requiring  each  side  of  the  equatKon  to   be 

multiplied  by  equal  quantities. 

1.  George  has  1  third  as  many  pears  as  apples,  and  the 
number  of  both  is  24.     How  many  has  he  of  each  ? 

Analysis. — If  x  represents  the  num- 
ber  of  apples,  then  -  will  represent  the 

number  of  pears,  and  x  +  -  will  equal 

24,  the  number  of  both.  Tbe  denomi- 
nator of  X  is  removed  by  multiplying 
each  term  on  both  sides  of  the  equation 
by  3.  (Ax.  6.)  The  result  is  3a;  +  x,  or 
4a;  =  72.  Hence,  cc=i8,  the  apples, 
and  18-7-3  =  6,  the  pears.     Hence, 

3 

107.  When  a  term  on  either  side  of  the  equation  has  a 
deiiominator,  that  denominator  is  removed  hy  multiplying 
every  term  on  hoth  sides  of  the  equation  hy  it.     (Ax.  4.) 

2.  What  number  is  that,  i  seventh  of  which  is  9  ? 

Ans.  6^, 

3.  What  number  is  that,  2  thirds  of  which  are  24  ? 

4.  A  man  being  asked  how  many  chickens  he  had, 
answered,  3  fourths  of  them  equal  18.     How  many  had  he  ? 

5.  What  number  is  that,  i  third  and  i  fourth  of  which 
are  21? 

Analysis. — If  x  represent  the  number,  then        j^^^     ^  __  ^^^ 

X        X 

will  -  +  -  =  21,  by  the  conditions.    Multiplying  x       X 

34  -+-=21 

each  term  on  both  sides  by  the  denominators  3  3        4 

and  4  separately,  we  have  /\x+2)X=-  252.    (Ax.  4.)  4^  "f"  3^  ^^  252 

Uniting  the  terms,  7a;  =  252,  and  x  —  36,  Ans.  ,' .       X  =    36 

Proof.    ^  of  36  =  12,  and  ^  of  36  =  9.    Now,  12  +  9  =  21. 

107.  When  a  quantity  on  cither  s'.de  .of  an  equation  has  a  denominator,  how  re- 
move it  ? 


MULTIPLICATION.  45 

6.  What  number  is  that,  2  thirds  of  which  exceed  i  half 
of  it  by  8  ? 

7.  A  general  lost  840  men  in  battle,  which  equaled 
3  sevenths  of  his  army.  Of  how  many  men  did  his  army 
consist  ? 

8.  If  3  eighths  of  a  yacht  are  worth  I360,  what  is  the 
l^hole  worth  ? 

A.C(y 

9.  If  --  equals  20,  to  what  is  x  equal  ? 

10.  If  —  is  equal  to  20,  to  what  is  x  equal  ? 

4 

11.  If  —  is  equal  to  24,  to  what  is  x  equal  ? 

AX 

12.  If  —  is  equal  to  28,  to  what  is  x  equal  ? 

13.  Henry  has  30  peaches,  which  are  5  sixths  the  number 
of  his  apples.     How  many  apples  has  he  ? 

14.  A  farmer  had  3  sevenths  as  many  cows  as  sheep,  and 
his  number  of  cows  was  30.  How  many  sheep  had  he  ? 
How  many  of  both  ? 

15.  Divide  28  pounds  into  two  parts,  such  that  one  may 
be  3  fourths  of  the  other. 

16.  A  lad  having  given  i  third  of  his  plums  to  one  school- 
mate, and  I  fourth  to  another,  had  10  left.  How  many  had 
he  at  first  ? 

17.  What  number  is  that,  i  third  and  i  sixth  of  which 
are  21? 

18.  What  number  is  that,  i   fourth  of  which  exceeds 

1  sixth  by  12  ? 

19.  Divide  36  into  two  parts,  such  that  one  may  be 

2  thirds  of  the  other  ? 

20.  One  of  my  apple  trees  bore  3  sevenths  as  many  apples 
as  the  other,  and  both  yielded  21  bushels.  How  many 
bushels  did  each  yield  ? 


CHAPTER     v. 
DIVISION. 

108.  Division  is  finding  how  many  times  one  quan- 
tity is  contained  in  another. 

The  Dividend  is  the  quantity  to  be  divided. 
The  Divisor  is  the  quantity  by  which  we  divide. 
The  Quotient  is  the  quantity  found  by  division. 
The  Remainder  is  a  part  of  the  dividend  left  after 
division. 

109.  Division  is  the  reverse  of  multiplication,  the  divi- 
dend answering  to  the  product^  the  divisor  to  one  of  the 
factors,  and  the  quotient  to  the  other, 

PRINCIPLES. 

110.  1°.  When  the  divisor  is  a  quantity  of  the  same  hind 
as  the  divide7id,  the  quotient  is  an  abstract  number. 

2°.  When  the  divisor  is  a  number,  the  qiiotient  is  a  quan- 
tity of  the  same  Tcind  as  the  dividend. 

3°.  The  product  of  the  divisor  and  quotient  is  equal  to  the 
dividend. 

4°.  Cancelling  a  factor  of  a  quantity,  divides  the  quantity 
by  that  factor. 

CASE    I. 

111.  To  Divide  a  Monomial  by  a  IVIonomial. 

I.  What  is  the  quotient  of  abed  divided  by  cd? 

Analysis.— The  divisor  cd  is  &  factor  of  the  divi-  operation. 

dend  ;  therefore,  if  we  cancel  this  factor,  the  other  cd  )  abcd 
factor  ab,  will  be  the  quotient.    (Prin.  4.)  Ans.  ab. 

108.  Define  division.  The  dividend.  Divisor.  Quotient.  Remainder.  109.  Oi 
what  is  division  the  reverse?  Explain,  no.  Name  Uie  first  principle.  The  second. 
Third.    Fourth. 


DIVISIOK.  4:7 

2.  What  is  the  quotient  of  i2>al)  divided  by  6a  ? 

Analysis.— DividiDg  the  coefficient  of  the  divi-  opbbation. 

dend  by  that  of  the  divisor,  and  cancelling  the  com-  6a  )  l?>ab 
mon  factor  a,  we  have  T8a6-j-6a  =  3&,  the  quotient  A^lS.  $b, 
required.    (Prin.  i.)    Hence,  the 

EuLE. — Divide  one  coefficient  by  the  other,  and  to  the  re^ 
suit  annex  the  quotient  of  the  literal  parts. 

Divide  the  following  quantities : 

(3-)  (4.)  (5-)  (6.) 

2c)  4abc  4b)  2obxy  8xy )  4oxy  16b)  S2ai 

(7.)  (8.)  (9-) 

gm  )  4sabm  2omn  )  6obcmn  24xy  )  g6mnxy 


SIGNS     OF    THE    QUOTIENT. 

112.  The  rule  for  the  signs  in  division  is  the  same  as 
ihat  in  multiplication.    That  is, 

If  the  divisor  and  dividend  have  lihe  signs,  the  sign  of  the 
quotient  will  be  + ;  if  unlike,  the  sign  of  the  quotient 
will  be  — . 

Thus,      +a  X  +b  =  +ab;  hence,  +ab -r-  +b  =  +a. 

—a  X  +b  =  —ab  ;  hence,  —ab  -r-  +b  =  —a. 

+  a  X  —b  =  —ab;  hence,  —ab  -. b  =  +a. 

—a  X  — &  =  +ab;  hence,  +ab  -i b  =  —a. 

Divide  the  following  quantities : 

10.  —  ^2abc  by  —  4ab.  Ans.  Sc. 

11.  iSabxhj—$z.  Ans. 

12.  2iabc  by  —  yab.  15.     4Sabc  by  —  Sac. 

13.  —  2Sbcd  by  —  4cd.  16.     6$bdfx  by  gbx. 

14.  $$cdm  by  7cm.  17.     —  'j2acgm  by  Scm. 

III.  How  divide  a  monomial  by  a  monomial?  112.  What  is  the  rale  for  the 
signs? 


48  DIVISlOITo 

113.  To  Divide  Powers  of  the  same  letter, 

1 8.  Let  it  be  required  to  divide  a^  by  a\ 

Analysis. — The  term  a^  —  aaaaa,  and  a^  —  aaa.  Rejecting  tho 
factors  aaa  from  the  dividend,  the  result  aa,  or  d^,  is  the  quotient. 
Subtracting  3,  the  index  of  the  divisor,  from  5,  the  index  of  the  divi- 
dend, leaves  2,  the  index  of  the  quotient.  That  is,  a^  -j-  a^  _  ^2 
(Arts.  31,  no.    Prin.  4.)    Hence,  the 

EuLE. — SuUraci  the  index  of  the  divisor  from  that  of  the 
dividend. 

Divide  the  following  quantities : 

19.  ^  by  6^.  22.  xyz^^hj  xyz^. 

20.  x^^  by  ^,  23.  i6ah^  by  ^ab. 

21.  ac^  by  ac^,  24.  6xy  by  3^2^ 

114.  The  preceding  principles  may  be  summed  up  in 
the  following 

Rule. — Divide  the  coefficient  of  the  dividend  hy  that  of  the 
divisor  ;  to  the  result  annex  the  quotient  of  the  literal  factors^ 
prefixing  the  proper  sign  and  giving  each  letter  its  proper 
exponent. 

Proof. — Multiply  the  divisor  and  quotient  together,  as  in 
arithmetic. 

Note. — If  the  letters  of  the  divisor  are  not  in  the  dividend,  the 
division  is  expressed  by  wanting  the  divisor  under  the  dividend,  in  the 
form  of  a  fraction. 

25.  What  is  the  quotient  of  ^x  divided  by  3?/?     Ans.  — • 

26.  —  24aWc^  -. 2>^ib.  32.  2>'^x^y^^  -^  dp?yz. 

27.  — 2i^x^y^^ -^  ^^y^'  33*  <)6aWc-^  i2ah. 

28.  $aW  -^  ah.  34.  84d^:i^y^  -7-  jd^xy. 

29.  —  jofiy^  -. xy.  35.  loSaix^  -i-  ga^a^. 

30.  a^b^c^ -T- aWc.  36.  i^2X*y!^ -r- iix^yzK 

31.  1 6aWc^  -7-  Sa^^c^.  37.  121  m^n^a^  -r-  1 1  mhia^. 

113.  ITow divide  powers  of  the  same  letter?  114.  Rule  for  division  of  moiia 
jnials  ?    Proof?    If  *^e  Jitters  of  the  divit-or  are  not  in  the  dividend,  what  i.-  done  ? 


DIVISIO.TST,  49 

CASE    II. 
115.  To  Divide  a  Polynottiial  by  a  Monomial. 

1.  Divide  ah  ■\-  ac  +  ad  by  a. 

Analysis.— Since  the  factor  a  enters  into  oPERATioif. 

each  term  of  the  dividend,  it  is  plain  that  a)  ab  -\r  dC  A-  ad 

each  term  of  the  dividend  must  be  divisible        Ans    h  4-    C  -i-     d 
by  this  factor.    Hence,  the 

EuLE. — Divide  each  term  of  the  dividend  ly  the  divisor, 
and  connect  the  results  hy  their  proper  signs. 

Note. — If  a  polynomial  which  contains  the  same  factor  in  every 
term,  be  divided  by  the  other  quantities  connected  by  their  signs,  the 
:iuotient  will  be  that  factor. 

Divide  the  following  quantities: 

2.  6a^  +  10^2  __  i^a  by  2a,  Ans,  sa^  4-  5«5  —  7. 

3.  40^  —  Sa^  +  i2«2  by  —  2«2.  Ans.  —  2^2  4.  4^5  _  6, 

4.  at/^  -i-  a(^  -{-  ad^  by  a. 

5.  isx^y  +  250:^  by  s^y. 

6.  6adc — 2a-i-Sah  by  2a, 

7.  —i6by^'{-4y^hj—Sy, 

8.  i4X^y  —  yxy^  by  —  yx^^ 

9.  xy^  -f  xz  —  xhj  X, 

10.  35fl5 -i- 28J— 42  by  — 7. 

11.  i$a^  —  15^2  by  5a. 

12.  i6x^ac  +  i2acd^  —  4xa^c  by  —  4CC. 

13.  4a*  —  20^2  +  Sab  by  4«. 

14.  3^5  -f  i^a^  —  2^a^)d  by  3flr5. 

15.  ZaV)c  —  xdatJ^c  —  2oabc^  by  ^abc, 

16.  6a;  {a  +  ^»)2  -+-  9a;2  {a  -{-  5)2  by  3ir. 

17.  ^S{^-y)-V2>o{x  —  y)hYS' 

18.  «a;2  (5  —  c)  —  aH  {b  —  c)  by  ax. 

19.   i8«4  (a  +  by  — 12«3  (^  _|.  ^,)2  by  6«2 («  4.  by. 

20.     a"+i  —  «"+2  4.  a«+3  by  «". 

115.  Eow  divide  a  polynomial  by  n  -lonomial  ? 


60  Divisioiy. 

CASE    III. 
116.  To  Divide  a  Polynomial  by  a  Polynomial. 

I.  Divide  a^  +  Za%  +  ^aV^  +  ^  by  a2  +  2ab  +  J*. 

Analysis.  —  For    conve-  opbbation. 

nience,  we  arrange  tlie  terms  a^ + sa^b  +  3«^  +  Z*^  a^ -|-  2rt J  4  5^ 
so  that  the  first  or  leading       a^^2a^+   aW  a  +  b  Quot 

.letter  of  the  divisor  shall  be  ^7~~      TTTTs 

the  first  letter  of  the  divi-  ^  0  +  2afr'+d 

dend.     The  powers  of  this  a^-{-2ai^+I)^ 

letter  should  be  arranged  in 

order,  both  in  the  divisor  and  dividend,  the  highest  power  standing 
Jirst,  the  neoct  highest  next,  and  so  on.  The  divisor  may  be  placed  on 
the  left  of  the  dividend,  or  on  the  right,  and  the  quotient  under  it,  at 
pleasure. 

Proceeding  as  in  arithmetic,  we  find  the  first  term  of  the  divisor  is 
contained  in  the  first  term  of  the  dividend  a  times.  Placing  the  a  in 
the  quotient  under  the  divisor,  we  multiply  the  whole  divisor  by  it, 
subtract  the  product,  and  to  the  remainder  bring  down  as  many  other 
terms  as  necessary  to  continue  the  operation.  Dividing  as  before,  a* 
is  contained  in  a%  +b  times.  Multiplying  the  divisor  by  +&  and 
subtracting  the  product,  the  dividend  is  exhausted  ;  therefore  a  +  &  ia 
the  quotient.     Hence,  the 

Rule. — I.  Arrange  the  divisor  and  dividend  according  to 
the  powers  of  one  of  their  letters ;  and  finding  how  many 
times  the  first  term  of  the  divisor  is  contained  in  the  first 
term  of  the  dividend,  place  the  result  in  the  quotient. 

IL  Multiply  the  whole  divisor  by  the  term  placed  in  the 
quotient ;  subtract  the  product  from  the  dividend,  and  tc  the 
remainder  bring  doivn  as  many  terms  of  the  dividend  as  the 
case  may  require. 

Repeat  the  operation  till  all  the  terms  of  the  dividend  are 
divided. 

Note. — If  there  is  a  remainder  after  all  the  terms  of  the  dividend 
ore  brought  down,  vlace  U  over  the  divisor,  and  annex  it  to  the  quotient 

ii6.  How  divide  a  polynomial  by  a  polynomial  ?  Jf  tba*o  is  a  remainder,  what  la 
done  with  it? 


PROBLEMS.  61 

2.  Divide  4^2  4-  ^ab  +  i^  hj  2a  +bo       Ans.  2a  +  b, 

3.  Divide  x^  +  2xy  -\-  y^  hj  x  -{-  t/, 

4.  Divide  a^  —  2ab  +  l^  by  a  —  b. 

5.  Divide  a^  --  sa^b  +  sab^  —  i,^  hy  a  ^  b. 

6.  Divide  ac  ■\-  be  -{-  ad  -\-  bd  hy  a  -{•  b, 

7.  Divide  ax  -{•  bx  —  ad  —  bd  by  a  +  b, 

8.  Divide  22^  -f  'jxy  4-  6^2  by  a;  +  2^ 
9/  Divide  a^  —  V^hya  +  b. 

10.  Divide  a?  —  y^  by  a;  —  y. 

11.  Divide  a^  —  l^  by  o  —  J. 

12.  Divide  6^3  4.  13^5^  4.  552  by  205  -f  35. 

13.  Divide  «2  __  ^  __  5  by  a  —  3. 

14.  Divide  a^  --  7,a^x  +  3aic2  _  ^^js  by  a  —  a; 

15.  Divide  6ar*  —  96  by  3a;  —  6. 

16.  Divide  x^  ■\-  "jx  -\-  10  by  x  -\-  2. 

17.  Divide  x^  —  5^?  +  6  by  x  —  3. 

18.  Divide  c^  —  2cx  -\-  7?  hy  c  —  a?. 

19.  Divide  a^  +  2ab  ■{- b^  hy  a  -{- h 

20.  Divide  22  («  —  Z>)2  by  1 1  (a  —  h). 


PROBLEMS. 

1.  A  father  being  asked  the  age  of  his  son,  replied,  My 
age  is  5  times  that  of  my  son,  lacking  4  years;  and  the 
sum  of  our  ages  is  5  6  years.     How  old  was  each  ? 

2.  John  and  Frank  have  60  marbles,  the  former  having 

3  times  as  many  as  the  latter.     How  many  has  each  ? 

3.  The  sum  of  two  numbers  is  72,  one  of  which  is  5  times 
the  other.     What  are  the  numbers  ? 

4.  A  man  divided  57  pears  between  two  girls,  giving  one 

4  times  as  many  as  the  other,  lacking  3.     How  many  did 
each  have  ? 

5.  Three  boys  counting  their  money,  found  they  had 
190  cents;  the  second  had  twice  as  many  cents  as  the  first, 
and  the  third  as  many  as  both  the  others,  plus  4  cents. 
How  many  cents  had  each  ? 


53  DIVISION 

6.  A  farmer  has  9  times  as  many  sheepas  cows,  and  the 
number  of  both  is  2oo„     How  many  of  each  ? 

7.  Divide  57  into  two  ?uch  parts  that  the  greater  shall  be 
3  times  the  less^  plus  3.    What  are  the  numbers  ? 

8.  Given  2X  -{-  4X  -\-  .4;  —  ^  =  60,  to  find  x, 

9.  A  and  B  are  35  miles  apart,  and  travel  toward  each, 
other,  A  at  the  rate  of  4  miles  an  hour,  and  B,  3  miles.  In 
bow  many  hours  will  they  meet  ? 

lOo  Given  a  -}-  $a  +  6a  +  2a  +  y  =  iig^  to  find  a. 

11.  Given  85  -f-  5^  -f  7^  —  10  =  130.,  to  find  b, 

12.  A  lad  having  60  cents,  bought  an  equal  number  of 
pears,  oranges^  and  bananas ;  the  pears  being  3  cents  apiece, 
the  oranges  4  cents,  and  the  bananas  5  cents=  How  many 
of  each  did  he  buy  ? 

13.  A  cistern  filled  with  water  has  two  faucets,  one  of 
which  will  empty  it  in  5  hours,  the  other  in  20  hours.  How 
long  will  it  take  both  to  empty  it  ? 

14.  Given  a;  +  -  =  45^  to  find  a?, 

15.  What  number  is  that,  to  ths  half  of  which  if  3  be 
added,  the  sum  will  be  8  ? 

160  Three  boys  have  42  marbles ;  B  has  twice  as  many  as 
As  and  C  three  times  as  many  as  A.    How  many  has  each  ? 

17.  If  A  has  2x  dollars,  and  B  twice  as  many  as  A,  and 
C  twice  as  many  as  B,  how  many  have  all  ? 

r8.  Divide  40  into  3  parts,  so  that  the  second  shall  be 
3  times  the  first,  and  the  third  shall  be  4  times  the  first. 

19.  A  man  divided  60  peaches  among  3  boys,  in  such  a 
manner  that  B  had  twice  as  many  as  A,  and  0  as  many  as 
A  and  B.     How  many  did  each  receive  ? 

20.  Divide  48  into  3  such  parts,  that  the  second  shall  be 
equal  to  twice  the  first,  and  the  third  to  the  sum  of  the  first 
and  second? 

21.  What  number  is  that,  to  three-fourths  of  which  if  5 
oe  added,  the  sum  will  be  23  ? 


OHAPTEE    VI. 
FACTORING. 

117.  Factors  are  quantities  which  multiplied  togethei 
produce  another  quantity.     (Art.  86.) 

118.  A  Composite  Quantity  is  the  product  of  two 
or  more  integral  factors,  each  of  which  is  greater  than  a 
unit. 

Thus,  3a,  5&,  also  x^y^,  are  composite  quantities. 

119.  Factoring  is  resolving  a  composite  quantity 
into  its  factors.     It  is  the  converse  of  multiplication. 

120.  An  Exact  Divisor  of  a  quantity  is  one  that  will 
divide  it  without  a  remainder.    Hence, 

Note. — The  Factors  of  a  quantity  are  always  exact  divisors  of  it, 
and  vice  versa. 

121.  A  Prime  Quantity  is  one  which  has  no  integral 
divisor,  except  itself  and  i. 

Thus,  5  and  7,  also  a  and  6,  are  prime  quantities.    Hence, 

Note. — The  least  divisor  of  a  composite  quantity  is  a  prime  factor. 

122.  Quantities  are  prime  to  each  other  when  they 
have  no  common  integral  divisor,  except  the  unit  i. 

Thus,  II  and  15,  also  a  and  6c,  are  prime  to  each  other. 

123.  A  Multiple  is  a  quantity  which  can  be  divided 
by  another  quantity  without  a  remainder.     Hence, 

A  multiple  is  a  product  of  two  or  more  factors. 

•117.  What  are  factors?  ii8.  A  composite  quantity?  119.  Wliat  is  factoring? 
120  An  exact  divisor?  121.  A  prime  quantity?  122.  When  prime  to  each  other? 
ii-3.  A  multioie? 


54  FACTORING. 


PRINCIPLES. 

124.  1^.  If  one  quantity  is  an  exact  divisor  of  another ^ 
the  former  is  also  an  exact  divisor  of  any  multiple  of  the 
latter. 

Thus,  3  is  a  divisor  of  6 ;  it  is  also  a  divisor  of  3  x  6,  of  5  x  6,  etc. 

2°.  If  a  quantity  is  an  exact  divisor  of  each  of  two  other 
quantities,  it  is  also  an  exact  divisor  of  their  sum,  their  dif- 
ference, or  their  product. 

Tims,  3  is  a  divisor  of  9  and  15,  respectively ;  it  is  also  a  divisor  of 
9+15,  or  24  ;  of  15—9,  or  6 ;  and  of  15  x  9,  or  135. 

3°.  A  composite  quantity  is  divisible  ly  each  of  its  prime 
factors,  by  the  product  of  two  or  more  of  them,  and  by  no 
other  quantity. 

Thus,  the  prime  factors  of  30  are  2,  3,  and  5.  Now  30  is  divisible 
hy  2,  by  3,  and  by  2  x  3  ;  by  2  x  5  ;  by  3  x  5  ;  by  2x3x5,  and  by  no 
other  number. 

CASE    I. 

125.  To  Find  the  Prime  Factors  of  Monomials. 
I.  What  are  tlie  prime  factors  of  i2«2j? 

Analysis. — The  coefficient  12  =  2x2x3,  and  a^h  —  adb.  There- 
fore the  prime  factors  of  i2a^6  are  2  x  2  x  '^aab.    Hence,  the 

KcTLE. — Find  the  prime  factors  of  the  numeral  coefficients, 
and  annex  to  them  the  given  letters,  taking  each  as  many 
times  as  there  are  units  in  its  exponent. 

Note. — In  monomials,  each  letter  is  a  factor.  Hence,  the  prime 
factors  of  literal  monomials  are  apparent  at  sight. 

Resolve  the  following  quantities  into  their  prime  factors : 

2.    is^f'  Ans.  zy^s^^y y y- 

3.  i2,d^b\  7.  i-jx^z. 

4.  lobo?]^,  8.  2$al^c3?, 

5.  $$aWc^,  9.  Tja^c^d. 

6.  2ixy^^.  10.  6$m^n^x. 

124.  Name  Principle  i.  Principle  2.  Principle  3.  125.  How  find  the  prime  fee- 
tors  of  monomials  ? 


FACTORING.  56 

CASE    II. 
126.  To  Factor  a  Polynomial. 

1.  Kesolve  ^a^h  +  ^ab  —  Sac  into  two  factors. 

Analysis. — By  inspection,  we  per-  operation. 

ceive  the   factor  2a  is  common  to  2a  )  /^a%  +  8fl^^  —  6ac 

each  term  ;  dividing  by  it,  the  quo-  2ah  +  ^h    —  3  c 

tient  2a&  +  4&-3/5  is  the  other  factor.         ^^^^  ^a  izah  +  4J  -  3c) 
For  convenience,  we  enclose  this  fac- 
tor in  a  parenthesis,  and  prefix  to  it  the  factor  2a,  as  a  coejfident. 

Proof. — The  factor  (206 + 4&— 3c)  x  2a=^a^l) + %ab—6ac.  Hence,  the 

EuLE. — Divide  the  polynomial  ly  the  greatest  common 
monomial  factor  ;  the  divisor  will  he  one  factor,  the  quotient 
the  other,     (Art.  115.) 

Note. — Any  common  factor,  or  the  product  of  any  two  or  more 
common  factors,  may  be  taken  as  a  divisor ;  but  the  result  will  very 
in  form  according  to  the  factors  employed.    (Ex.  2.) 

2.  Eesolve  a^  +  alt^  into  two  factors,  one  of  which  shall 
be  a  monomial.     Ans.  db  (a  +  ^),  a  (ah -{-¥),  or  h  (a^  -f  ah), 

3.  Factor  a  +  ah  +  ac.  Ans,  a  (i  -^  h  -\- c), 

4.  Factor  by  -\-  he  -\-  t^x, 

5.  Factor  2ax  •\-  2ay  —  4az, 

6.  Factor  ^hcx  —  Ghcx  —  ^ah(k 

7.  Factor  Mmn  —  2^dm^ 

8.  Factor  35^7/2  -f-  14^0:. 

9.  Factor  2^hdx^^^dmy, 

10.  Factor  6a^  -f  ga^c. 

11.  Factor  2iao[^y  -f-  35«a;y. 

12.  Factor  25  +  152:2  —  200^2^. 

13.  Factor  x  -\-  x^  -\-  ^, 

14.  Factor  3a;  -f  6  —  9^. 

15.  Factor  19^52; —19^5, 


126.  How  factor  a  polynomial  ? 


56  rACTORiiir®. 

CASE    III. 
127.  To  Resolve  a  Trinomial  into  two  equal  Binomial  Factors. 

1.  Resolve  x^  +  2xy  -f  y"^  into  two  equal  binomial  factors. 

Analysis. — Since  tlie  square  of   a        *  operation. 

quantity  is  the  product  of  two  equal  /^xi  ==:  X,      V^  =  2/> 

factors  (Art.  30),  it  follows  tliat  the  ^        c^  2 

square  root  of  a  quantity  is  one  of  the  "  '        "  '"  •2' 

two  equal  factors  which  produce  it.  (^H"^)  (^  +  2/)?   Ans* 

(Art.  32.)    Therefore  the  square  root  of 

Q?  is  ar,  that  of  y^  is  y.     And  since  the  middle  term  2Ty  is  twice  the 

product  of  these  two  terms,  x^->r2xy+y'^  must  be  the  square  of  the 

binomial  x+y.    Consequently,  x  +  y  is  one  of  the  two  equal  binomial 

factors. 

2.  Resolve  x^  —  2xy  -f-  y"^  into  two  equal  binomial  factors. 

ANAiiTSis. — Reasoning  as  before,  the  operation. 

quantity  x^—2Ty+y^  is  the  square  of  ^x^  z=x,      V^  =  ^ 

the  residual  x—y.    Therefore,  the  two  ^         „ j_     2 

equal  factors  must  be  x—y  and  x—y.  '  '                   if  '^  if 

Hence,  the  (^— ^)  (^  —  y)>  ^^S. 

Rule. — Find  the  square  root  of  each  of  the  square  terms, 
and  connect  these  roots  hy  the  sign  of  the  middle  term. 

Note. — A  trinomial,  in  order  to  be  resolved  into  equal  binomial 
factors,  must  have  two  of  its  terms  squares,  and  the  other  term  timce 
the  product  of  their  square  roots.    (Art.  loi.) 

Resolve  the  following  into  two  equal  binomials: 


3- 

^2  +  2a'b  +  J2. 

9. 

f+2y+i. 

4. 

x^—2xy-\-  y\ 

10. 

I  _  2C2  +  C^. 

5. 

m^  -j-  4mn  +  4^2. 

II. 

^2m  ^   2^myn  ^  y^^ 

6. 

i6a^  +  8«  +  I. 

12. 

4^2"  _  4a»  4-  I. 

7- 

49  +  70  +  25. 

13. 

a^  +  2aW  +  IK 

8. 

4^2  _  i2ah  +  9J2. 

14. 

a^x^  +  2ax^y  +  y\ 

197.  How  resolve  a  trinomial  into  equal  binomial  fectors  f 


PACTOBING.  67 


CASE    IV, 


2.28.  To  Factor  a  Binomial  consisting  of  the  Difference  of 
two  Squares. 

I.  Resolve  ^a^  —  9J2  into  two  binomial  factors. 

Analysis.— Both  of  these  terms  operation. 

are  squares;  the  root  of  the  first  is  ^ ^0?  =  2fl5 

2a,  that  of  the  second  is  36.    But  the  /~w 7 

difference  of  the  squares  of  two  quan-  ^ 

tities  is  equal  to  the  product  ot  their  *  '     ^^         9^ 

sum  and  difference.   (Art.  103.)  Now  (2«4-35)  (2a— 3^),  ^W5. 

the  sum  of  these  two  quantities  is 

2a  +  3&,  and  the  difference    is    2a  —  3& ;    therefore,    j\a?  —  96^  = 

(2a  +  3&)(2a— 3&).    Hence,  the 

Rule. — Find  the  square  root  of  each  term.     The  sum  of 
these  roots  will  he  one  factor,  and  their  difference  the  other. 

Note. — This  rule  is  one  of  the  numerous  applications  of  the  for- 
mula contained  in  Art.  103. 

2.  Resolve  a^  —  x^  into  two  binomial  factors. 

3.  Resolve  ^x^  —  i6y^  into  two  binomial  factors. 

4.  Resolve  y^  —  4  into  two  binomial  factors. 

5.  Resolve  g  —  x^  into  two  binomial  factors. 

6.  Resolve  a^  —  i  into  two  binomial  factors. 

7.  Resolve  i  —  ^  into  two  binomial  factors. 

8.  Resolve  25^^  —  16^2  into  two  binomial  factors. 

9.  Resolve  ^x^  —  y^  into  two  binomial  factors. 
10.  Resolve  1  —  16^2  into  two  binomial  factors. 

II.  Resolve  25  —  i  into  two  binomial  factors. 

12.  Resolve  x^  —  y*  into  two  binomial  factors. 

13.  Resolve  a^x^  —  b^y^  into  two  binomial  factors. 

14.  Resolve  m^  —  #  into  two  binomial  factors. 

15.  Resolve  a^'""  —  h'^"  into  two  binomial  factors. 

xaS,  How  fector  a  binomial  consisting  of  the  difference  of  two  squares  ? 


58  1^  AGIO  KING. 


CASE    V. 

129.  Various  classes  of  examples  of  higher  powers  may  be 
factored  by  means  of  the  following 

PRINCIPLES. 

1°.  The  difference  of  any  tivo  powers  of  tlie  same  degree  is 
divisible  by  the  difference  of  their  roots. 

Thus,    (x^—f)  -J-  {x—y)  —  x+y. 

{^—f)  -4-  {x-y)  =  a^+xy+^. 
(oe*—y^)  H-  {x—y)  —  a^  +  x^y+xy^+y\ 
(a^— y5)  -r-  (x—y)  =  x^  +  Q^y  +  x'^y'^  +  xy^+y^. 

2°.  The  differ e7ice  of  two  even  poivers  of  the  same  degree  is 
divisible  by  the  sum  of  their  roots. 

Thus,    {Q?—y^)  -i-  (x+y)  —  x—y. 

(aj4_2^)  ^  (x+y)  =  x^—x^y+xy'^—y^. 

(afi—y^)  4-  (x+y)  =  x^—x^y  +  a^y^—x^y^  +  xy*—^, 

3°.  TJie  Slim  of  two  odd  poicers  of  the  same  degree  is  divi- 
sible by  the  sum  of  their  roots. 

Thus,    i^+y^)  -^  {x+y)  =  o^-xy+yK 

(a^+y^)  -r-  (x+y)  =  xl^—a^y  +  x-y'^—xy^+y*. 

(x'^+y'')  ■+■  (x+y)  —  afi—x^y+xY—a^y^+xY—xy^+y\  etc. 

Note.— The  indices  and  signs  of  the  quotient  follow  regular  laws : 
I  St.  The  index  of  the  first  letter  regularly  deoreoMS  by  i,  while  that 
of  the  following  letter  increases  by  i. 

2d.  When  the  difference  of  two  powers  is  divided  by  the  difference 
of  their  roots,  the  signs  of  all  the  terms  in  the  quotient  are  pltis.  When 
their  sum  or  difference  is  divided  by  the  sum  of  their  roots,  the  odd 
terms  of  the  quotient  are  plus,  and  the  even  terms  minus. 

1^  If  the  principles  and  examples  of  this  Case  are  deemed  too 
difficult  for  beginners,  they  may  be  deferred  until  the  Binomial 
Theorem  is  explained.    (Arts.  268-270.) 

129.  Kecite  Prin.  i.  Prin.  2.  Prin.  3.  Note.  What  is  the  index  of  the  first  let 
ter  f    Of  the  following  letter  ?    What  is  said  of  the  si^uB  ? 


FACTORING.  59 

130.  To  Factor  the  Difference  of  any  two  Powers  of  the 
same  Degree. 

I.  Eesolve  a?  —  y'^  into  two  factors. 

Solution.— The  binomial  (a^— y^)  -^  {x—y)  =  x^  +  xi/+^.  .*.  x—y 
md  x^  +  xy+y^  are  the  factors.    (Prin.  i.)    Hence,  the 

'RuL^.— Divide  the  difference  of  the  powers  hy  the  differ- 
ence of  the  roots;  the  divisor  will  he  one  factor,  the  quotient 
the  other, 

Eesolve  the  following  into  two  factors : 
3'     ^  —  y^'  5.     I  — 


131.  To  Factor  the  Difference  of  two  even  Powers  of  the 

same  Degree. 

6.  Eesolve  a^  —  Z>*  into  two  factors. 

Solution.— By  Prin.  2,a^—¥ia  divisible  hy  a  +  b.  Thus,  (a*— 6^) 
^  (a  +  b)  =  a^—a^b  +  c^—b^,  the  divisor  being  one  factor,  the  quotient 
the  other.    Hence,  the 

EuLE. — Divide  the  difference  of  the  given  powers  ly  the 
sum  of  their  roots ;  the  divisor  ivill  he  one  factor,  the  quo- 
tient the  other,     (Art.  129,  Prin.  2.) 

Eesolve  the  following  quantities  into  two  factors : 

7.  J2  _  x\  10.     iC^  —  1. 

8.  d^  —  ^,  II.     1—^6, 

9.  a^  —  h^,  12.     a^  —  I. 

132.  To  Factor  the  Sum  of  two  odd  Powers  of  the  same 

Degree. 

13.  Eesolve  a}  +  h^  into  two  factors. 

Solution. — Dividing  a^  +  V^  by  a  +  b,  the  factors  are  a  +  b  and 
a^—ab+y"*      (Prin.  3.)    Hence,  the 

EuLE. — Divide  the  sum  of  the  powers  hy  the  sum  of  the 
roots;  the  divisor  and  quotient  are  the  factors. 

130.  How  factor  the  diflference  of  any  two  powers  of  the  same  degree  ?  131.  How 
factor  the  difference  of  two  even  powers  of  the  same  degree  ?  132.  The  sum  of  two 
odd  powers  of  the  same  degree  t 


60  FACTOEIKG. 

Resolve  the  following  quantities  into  two  factors : 

14.  a^-{-f.  17.     i  +  y\ 

15.  a^  +  I.  18.     I  +  a\ 

16.  a^  +  I.  19.     I  +  W. 

133.  It  will  be  observed  that  in  the  preceding  examples 

of  this  Case,  binomials  have  been  resolved  into  tivo  factors. 

These  factors  may  or  may  not  be  prime  factors. 

Thus,  in  Ex.  6,  d^-h^  =  {a  +  'b){a^-a'^h-\- (0)^-1)%  But  the  /'actor 
{a^—a?h-\-ah'^—¥')  is  a  composite  quantity  =  {a—b)(a^  +  ¥). 

134.  When  a  binomial  is  to  be  resolved  into  prime  fac- 
tors, it  should  first  be  resolved  into  two  factors,  on<>  of 
which  is  prhne ;  then  the  composite  factor  should  be 
treated  in  like  manner. 

20.  Let  it  be  required  to  find  the  prime  factors  of  «^  -  ¥. 

Solution.— The    ^/a^  =  a\    and    /y/ft^^ft^      (Art.  128.) 

Now  a^-¥-  =  (a2-&2) (^2  +  jf.y    But  a^-h^  =  («+&)  {a-V).  (Art.  i^j.) 

Therefore  the  prime  factors  of  a^—¥  are  {a^  +  ¥){a  +  l){a—h). 

Resolve  the  following  quantities  into  their  prime  factors  ' 

21.  «^— I.  Arts.  {dJ^  ■\-i){a-\- i){a—i). 

22.  i—f.  Ans.  (i+y^){i-{-y){i-yy 

23.  ofi  —  y^. 

Ans.  {x^  -xy  +  y^)  {a^  +  ccy  +  y^)  {x  •\-y){x  —  y). 

24.  x^  —  2x^y^  4-  y^. 

Ans,  {x^  -  y^y  =  {x  +  y)  {x -h  y)  (^  -  «/)  (^  ~  V)- 

25.  a^ —  I. 

Ans.  (x  4-  i)  {x-  i)  {x^  -\-x^-i){xi-x-\-  i). 

26.  a«  +  2a%^  +  2'^. 

^?^5.  {a  -\-h){a  +  I)  {a^  -  a2>  +  ^)  {a^  -  ah -]r  ^), 

27.  «2  _^  9a  +  18.  Ans.  {a  +  6)  (rt  +  3}. 

28.  4^2  ~  i2«&  +  9&2.  ^ws.  (2a  —  2fi)  (20    -  3^) 

(See  Appendix,  p.  284.) 


OHAPTEE    YII 
DIVISORS    AND    MULTIPLES. 

135.  A  Common  I>ivisor  is  one  that  will  divide  two 
or  more  quantities  without  a  remainder. 

136.  Commensurable  Quantities  are  those  which 
haye  a  common  divisor. 

Thus,  aW  and  dbc  are  commensurable  by  ah. 

137.  Incommensurable    Quantities    are    those 
which  have  no  common  divisor.     (Art.  122  ) 

Tims,  ab  and  xyz  are  incommensurabla. 

138.  To  Find  a  Common  Divisor  of  two  or  more  Quantities. 

1.  Find  a  common  divisor  of  abx,  act/,  and  adz. 
Analysis. — Resolving  the  given  quanti-  operation. 

ties  into  factors,  we  perceive  the  factor  a,  is  abx  =  a  xb  XX 

common  to  each  quantity,  and  is  therefore  a  act/  =  axcXV 

common  divisor  of  tbem.    (Art.  119,)  Hence,  ^^^i  ^  axdxz 

*^^  A71S.  a. 

Rule. — Resolve  each  of  the  given  quantities  into  factors, 
one  of  ivhich  is  common  to  all. 

Find  a  common  divisor  of  the  following  quantities : 

2.  -^alcd  and  gahm.  Ans.  ^ah. 

3.  x^yz  and  lahx.  6.  2ax,  dlx,  14CX. 

4.  a%  bed,  ah^xy.  7.  35^^,  "jm"^,  42m^x. 

5.  2adc,  acx%  a^cy,  8.  24a^,   i2al>^,  6aW. 

135.  What  is  a  common  divisor  ?    136.  Commensurable  quantities  ?    137.  Incom- 
mensunible  quantities  ?    138.  How  find  a  common  divisor  of  two  or  more  quantities  ? 


0x5  DIYIS0R8. 

139.  The  Greatest  Common  Divisor  of  two  or 

more  quantities  is  the  greatest  quantity  that  will   divide 
each  of  them  without  a  remainder. 

Notes. — i.  A  common  divisor  of  two  or  more  quantities  is  always  a 
common  foMor  of  those  quantities,  and  the  g.  c.  fZ.*  is  their  greatest 
common  factor. 

2.  A  common  divisor  is  often  called  a  common  measure,  and  the 
greatest  common  divisor,  the  greatest  comm/m  measure. 


PRINCIPLES. 

140.  1°.  TJie  greatest  common  divisor  of  two  or  more 
quantities  is  the  product  of  all  their  common  prime  factors. 

2°.  A  commoji  divisor  of  tioo  quantities  is  7iot  altered  ty 
multiplying  or  dividing  either  of  them  hy  any  factor  not 
found  in  the  other. 

Thus,  3  is  a  common  divisor  of  i8  and  6  ;  it  is  also  a  common  divisor 
of  1 8,  and  of  (6  x  5)  or  30. 

3°.   The  signs  of  a  2^olynomial  may  he  changed  hy  divid- 
ing it  hy  —  I. 
Thus,  (— 3a  +  4&— 5c)h 1  =  3a— 46  + 5c.    (Art.  112.)    Hence, 

4°.  Tlie  signs  of  the  divisor,  or  of  the  dividend,  or  of  hoth, 
may  he  changed  without  changing  the  common  divisor. 

141.  To  Find  the  Greatest  Common  Divisor  of  Monomials  by 

JPrune  Factors, 

I.  What  is  the  g,  c,  d,  of  35aca;,  2Sahc,  and  2iay? 

Analysis.  —  Resolving   the  operation. 

given  quantities  into  their  prime  3SCICX  :=  <^XT  XaXCXX 

factors,  7  and  a  only,  are  com-  ^SaJc  =  2X2X7X«X*XC 
mon  to  each ;    therefore  their 

product  7  X  a,  IS  the  g,c,a,re-  ^         -»      /  ./ 

quired.    (Prin.  i.)  •'•     7Xfl5  =  7«-      ^ns. 

139*  What  is  the  greatest  common  divipor  of  two  or  more  quantities  ?  Note  i. 
What  is  true  of  a  common  divisor  of  two  or  more  quantities?  Of  the  g.  c.  d.? 
140.  Name  Principle  i.    Principle  2.    Principle  3. 

*  The  initials  g,  c,  d,  are  usod  for  the  greatest  common  divisor. 


jivisoRS.  63 

2.  Find  the  g,  c,  d.  of  4a^b%  loaW,  and  i^abdx. 

Analysis. — Resolving  these  quan-  operation. 

titles  into  their  prime  factors,  the  ^aWc  =  2  x  2  X  oaahhc 

factor  2  is  common  to  the  coefficients;  locfil^  =  2  X  5  X  CLobbb 

also,  a  and  6  are  common  to  the  lit-  ^o^aMx  =  2  X  7  X  obdx 

eral  parts      Now  multiplying  these  ^^^^   2^a^h  =  2ab 
common  factors  together,  we  have 
ly.ay.'b  —  2ab,  which  is  the  g.  c, d.  required.    (Prin.  i.)    Hence,  the 

Rule. — Resolve  the  given  quantities  into  their  prime  fac- 
tors ;  and  the  product  of  the  factors  common  to  all,  will  he 
the  greatest  common  divisor.     (Prin.  i.) 

Note. — In  finding  the  common  prime  factors  of  the  literal  part, 
give  each  letter  the  least  exponent  it  has  in  either  of  the  quantities 

3.  Find  the  g,  c.  d,  of  6aV  and  ^dbc, 

4.  Of  i6a^xy  and  i^acx^y. 

5.  Of  i2aWa^^  and  i6a^x^zK 

6.  Of  daV^x^^,  i2a^xh^,  ana  i2>a^xh\ 

142.  To  Find  the  Greatest  Common  Divisor  of  Quantities  by 
Continued  Division, 

I.  Required  the  greatest  common  divisor  of  30a;  and  422;. 

Analysis. — If  we  divide  the  greater  operation. 

quantity  by  the  less,  the  quotient  is  i,         30a; )  42:?;  (  i 

and  12a;  remainder.     Next,  dividing  the  302; 

first  divis'or  30a;,  by  the  first  remainder  \  / 

^1,  J.-     \  ■  J  XI  -J  I2iC)302;(2 

12a;,  the  quotient  is  2  and  the  remainder  ^  ^      ^ 

6x.    Again,  dividing  the  second  divisor  ^4^ 

by  the  second  remainder,  the  quotient  6x  )  1 2:?;  (  2 

is  2  and  no  remainder.    The  last  divisor, 

tx,  is  the  ff,  c,  d. 


12a; 


Demonstration. — Two  points  are  required  to  be  proved  : 
ist.  That  6x  is  a  common  divisor  of  the  given  quantities. 
2d.  That  6a!  is  their  greatest  common  divisor. 

First.  We  are  to  prove  that  6x  is  a  common  divisor  of  30a;  and  42a;. 
By  the  last  division,  6a;  is  contained  in  123*,  2  times.    Now  as  6a;  k  a 

141.  How  find  the  g.c.  d.  of  monomials  by  prime  factors?    Note.  In  finding  the 
prime  factors  of  the  literal  part,  what  exponents  are  given  ? 


64  DIVISORS, 

divisor  of  i2aJ,  it  is  also  a  divisor  of  the  product  of  i2aj  into  2,  or  24a;. 
(Art.  124,  Prin.  i.)  Next,  since  6x  is  a  divisor  of  itself  and  24a*,  it 
must  be  a  divisor  of  the  sum  of  tx-\-2^,  or  30a!,  which  is  the  smaller 
quantity.  For  the  same  reason,  since  ()X  is  a  divisor  of  12a;  and  30a;,  it 
must  also  be  a  divisor  of  the  sum  of  12a; +  302;,  or  42a;,  which  is  the 
larger  quantity.    Hence,  tx  is  a  common  divisor  of  30a;  and  42a;. 

Second.  We  are  to  prove  that  6x  is  the  greatest  common  divisor  of 
30a;  and  42a?. 

If  the  greatest  common  divisor  is  not  bx,  it  must  be  either  greater 
or  less  than  bx.  But  we  have  shown  that  tx  is  a  common  divisor  of 
the  given  quantities  ;  therefore,  no  quantity  less  than  tx  can  be  the 
greatest  common  divisor  of  them.  The  assumed  quantity  must  there- 
fore be  greater  than  bx.  By  supposition,  this  assumed  quantity  is  a 
divisor  of  30a?  and  42a! ;  hence,  it  must  be  a  divisor  of  their  difference, 
42aj— 30a;,  or  12a;.  And  as  it  is  a  divisor  of  12a;,  it  must  also  divide  the 
product  of  12a;  into  2,  or  24a;. 

Again,  since  the  assumed  quantity  is  a  divisor  of  3oaJ  and  24a;,  it 
must  also  be  a  divisor  of  their  difference,  which  is  bx ;  that  is,  a  greater 
quantity  will  divide  a  less  without  a  remainder,  which  is  impossible. 
Therefore,  6a!  must  be  the  greatest  common  divisor  of  3oaj  and  42aJ, 
the  second  point  to  be  proved.    Hence,  the 

B,XJL^.— Divide  the  greater  quantity  by  the  less,  then  divide 
the  first  divisor  dy  the  first  remainder,  the  second  divisor  by 
the  second  remainder,  and  so  on,  till  there  is  no  remainder. 
The  last  divisor  will  be  the  greatest  common  divisor. 

Note. — If  there  are  more  than  two  quantities,  fifid  the  g.  c»  d, 
of  the  smaller  two,  then  of  this  common  divisor  and  a  third  quantity, 
and  so  on  with  all  the  quantities. 

2.  What  is  the  g,  c,  d.  of  48^5,  72  a,  and  io8a? 

Suggestion. — The  </,  c,  d,  of  48a  and  72a  is  24a  ;  and  that  of  24a 
and  loSa  is  12a.     Therefore,  12a  is  the  (/.  c,  d,  required. 

142, «.  The  Greatest  Common  Divisor  of  Poly- 
nomials is  found  by  the  above  rule,  as  illustrated  in  the 
following  examples: 

142.  Show  upon  the  blackboard  the  truth  of  this  rule?  r.-p  a  How  find  the 
g.  ^.  d.  of  polynomials  ? 


DIVISORS 


65 


3.  What  is  the  g.  c.  d.  of  /^a^  —  2ia2  4.  15a  4.  20    and 

<j3  __  6a  +  8  ? 


OPKBATION. 


^a^  —  2ia^  +  150^  +20     a^  —  6«  +  8    istdiviBor. 


\a^  —  24^2  4.  32« 


4^  +  3 


zst  quotient. 


-f    3a^  —  1 7a  +  20 
f     3^2  _  i8a  -f  24 


a«  —  6«  +  8 

a^  —  /^a 

—  2a  +  8 

—  2a  +  8 


a  —     4      'Pt  remainder  and  2d  divisor. 


a  —     2      ad  quotient. 

Ans.  a  —  4. 


Analysis. — Dividing  the  greater  quantity  by  the  less,  the  remain- 
der is  a— 4.  Again,  dividing  the  first  divisor  by  the  first  remainder, 
the  quotient  is  a— 2,  and  no  remainder.  The  last  divisor,  a— 4,  is  the 
greatest  common  divisor. 

143.  It  is  sometimes  necessary,  in  order  to  avoid  frac- 
tions, to  introduce  a  factor  into  one  or  both  the  given 
quantities,  or  to  cancel  one  before  finding  the  greatest  com- 
mon divisor. 

It  is  also  sometimes  necessary  to  change  the  signs  of 
the  divisor  or  dividend,  or  of  both.     (Art.  140,  Prin.  4.) 

4.  What  is  the  g,  c.  d,ofa^—  2xy  +  y^  and  01?  —  y^"^ 


Analysis.— Dividing  the 
greater  by  the  less,  the  first 
remainder  is  —  2xy  +  2y^. 
Cancelling  from  it  the  com- 
mon factor  2y,  we  have  for 
the  second  divisor  —x  +  y. 
Changing  the  signs,  it  be- 
comes a;— y.  (Art.  140,  Prin.  4.) 
Dividing  as  before,  the  quo- 
tient \^  x  +  y,  and  no  remainder, 
common  divisor. 


OPKBATION 

0^ — 2Xy-\-    y^ 


7?  —    ^2 

2y)  —2xy+2f' 


Divisor,      — ^  +  y 

or,  ^—y 

Quotient,         ^-\-y 


X^ — y^    Divisor, 
I       Quotient. 


a? — y"^ 


■t 


X?- 


The  last  divisor,  x—y,  is  the  greatest 


143.  How  does  it  affect  the  g.  c.  d.  if  a  factor  is  introduced  into  eitlier  orbotli  the 
given  quantities  ?    How  if  one  is  cancelled  ?    What  is  true  of  the  si^jne  ? 


66 


DIVISORS. 


5.  What    is    the  g.  c,  d.  of    4^3  —  6a;2  —  4a;  +  3    and 

2X^  +  X^  +  X—1? 


dividend,   4^  —  6x^  —  4X  +  S 

43:^  -\-   2X^  -{-  2X  —  2 

2d  divisor,         —  8,7^  —  6a;  -f  5 
2d  quotient, 


OPERATION. 

2C(^  -^  X^  -^  X  —  I      ist  divisor. 
2  igt  quotient. 


—  X 


3d  dividend,  —  Sx^  —     6x  -{-     5 
—  Sx^  +  S^X  —  16 


4th  divisor, 
4tli  quotient. 


—  21  )  —  42a;  +21 
,  2ir  —   I 


—  a;  +  4 


8a.'^  +  4X^  -{-  4X  —  4  2d  dividend 

8'Ji^  +  6a;2  —  5ic 

—  22:^  -i-  9a;  —  4    3d  divisor. 
4  3d  quotient. 

—  23^^+93;  —  4        4th  dividend. 
~  2X^  -{-      X 


2X  —  I  is  the  g,  c,  d. 


8a;  — 4 
8a;  —  4 


Analysis. — Dividing  the  greater  by  the  less,  the  first  term  of  the 
first  remainder,  —  Sa*-,  is  not  contained  in  2a:^>  the  first  term  of  the 
second  dividend.  We  therefore  multiply  this  dividend  by  4,  and  it 
becomes  Sx^  +  4X^  +  4X— 4,  and  dividing  this  by  the  second  divisor,  the 
second  remainder  is  —22;'^ +  90?— 4.  Dividing  the  preceding  divisor  by 
this  remainder,  we  see  that  the  third  remainder,  —422;  + 21,  is  not 
contained  in  the  next  dividend.  Cancelling  the  factor  —21,  the  fourth 
divisor  becomes  2X  —  i,  the  greatest  common  divisor  required. 

Find  the  g,  c.  d.  of  the  following  quantities: 

6.  a^  —  y^  and  a^  —  2x2/  +  ^. 

7.  «3  _f_  ^z  an^  ^2  _j_  2ab  +  ^. 

8.  h^  —  4  and  J2  ^  45  +  4. 

9.  a:^  —  9  and  x^  -{-  6x  -\-  9. 

10.  a^  —  3(1  +  2  and  a^  —  a  —  2. 

11.  «^  +  3^2  -f  4flj  -|-  12  and  a^  -f  4^2  _{-  4^5  _{-.  3, 

12.  a^  -{-  I  and  x^  +  mx^  -f  772a;  +  1. 

13.  d^  —  Ifi  and  a^  —  W, 

14.  fl^2  _  2^j  _j_  4^2  and  c^  —  a^h  +  3fl!j2  _  ^js 

15.  3^  —  102;^  +  15^  +  8  and  x^  —  2a;^  —  6ar^  +  4a:2 
4-  13a;  -f  6. 


MULTIPLES.  ;37 


MULTIPLES. 

144.  A  Multiple  is  a  quantity  which  can  be  divided 
by  another  quantity  without  a  remainder.     (Art.  123.) 

145.  A  Coimnon  Multijde  is  a  quantity  which  can 
be  divided  by  two  or  more  quantities  without  a  remainder. 

Thus,  1 8a  is  a  common  multiple  of  2,  3,  6,  and  9. 

146.  The  Least  Common  Multiple  of  two  or 

more  quantities  is  the  least  quantity  that  can  be  divided  by 
each  of  them  without  a  remainder. 

Thus,  21  is  the  least  common  multiple  of  3  and  7 ;  30  is  the  least 
common  multiple  of  2,  3,  and  5. 


PRINCIPLES. 

147.  1°.  ^  multiple  of  a  quantity  must  contain  all  the 
prime  factors  of  that  quantity. 

Thus,  18  is  a  multiple  of  6,  and  contains  the  prime  factors  of  6, 
which  are  2  and  3. 

2°.  A  common  multiple  of  two  or  more  quantities  must 
contain  all  the  prime  factors  of  each  of  the  given  quantities. 

Thus,  42,  a  common  multiple  of  14  and  21,  contains  all  the  prime 
factors  of  those  quantities  ;  viz.,  2,  3,  and  7. 

3°.  The  least  com7non  multiple  of  tiuo  or  more  quantities 
is  the  least  quantity  which  contains  all  their  prime  factors^ 
each  factor  heing  tahen  the  greatest  numderof  times  it  occurs 
in  either  of  the  given  quantities. 

Thus,  30  is  the  least  common  multiple  of  6  and  to,  and  contains  all 
the  prime  factors  of  these  quantities  ;  viz.,  2,  3,  and  5. 

144.  What  is  a  multiple  ?  145,  A  common  multiple  ?  146.  The  least  common 
multiple  ?    147.  Name  Principle  i.    Principle  2.    Principle  3. 


68  MULTIPLES. 

148.  To  Find  the  Least  Common   Multiple  of  Monomials  by 
Prime  Factors, 

1.  Find  the  I,  cm,*  of  i5«V,  ^l^cx,  and  9&A 

Analysis. — The   prime  factors  operation. 

of  the  coefficients  are  5,  3,  and  3.  l^^a'^OC^  =  3X5X«*Xa? 

The  prime  factors  of  the  letter  a  t^W-cx  =1  ^xl^  XCXX 

are  a,  a,  a,  a,  which  are  denoted  by  qbc^z  =  ^X^X^X<^X^ 

a*.     In  like  manner,  the  prime  fac-  j^^^^  A^a^l^C^Z. 

tors  of  X  are  denoted  by  a;',  those  of 

b  by  b'^,  and  those  of  c  by  c^ ;  z  is  prime.  Taking  each  of  these  factors 
the  greatest  number  of  times  it  occurs  in  either  of  the  given  quanti- 
ties, the  product,  45a^&-c%-2,  is  the  I,  c,  in,  required.  (Art.  147,  Prin.  2.) 
Hence,  the 

Rule. — Resolve  the  quantities  into  their  prime  factors  ; 
multiply  these  factors  together,  tahing  each  the  greatest  num- 
her  of  times  it  occurs  in  either  of  the  given  quantities.  Tlie 
product  is  the  I,  c,  m,  required. 

Or,  Find  the  least  common  multiple  of  the  coefficieyits,  and 
annex  to  it  all  the  letters,  giving  ecLch  letter  the  exponent  of 
its  highest  poiver  in  either  of  the  quantities. 

Note. — Tn  finding  the  I.  c,  in,  of  algebraic  quantities,  it  is  often 
more  expeditious  to  arrange  them  in  a  horizontal  line,  then  divide, 
«tc.,  as  in  arithmetic. 

Eequired  the  I,  c,  Vfl,  of  the  following  quantities : 

2.  ^a\  i2a^A  and  24^2:2^.  Ans.  yza^x^g, 

3.  ^ab^  2Sbc^,  and  ^6a^M, 

4.  iSoc^'ifz,  2oy%  and  Sxyz^. 

5.  isa^l^c,  ga¥c%  and  iSa^c^. 

6.  2Sab\  i4«2J*,  35«^^,  and  42a*h. 

7.  2ix^yh^,  35^y^z^,  and  d^xyH. 

8.  ^mVy,  i2mhiy'^,  2i>Yi^  T^mn^y^. 

148.  How  find  the  I.  c.  tn.  of  monomials  by  prime  factors  ?  What  other  method 
*  The  initials  I,  c,  tn,  are  used  for  the  least  common  multiple. 


MULTIPLES.  69 

149.  To  Find  the  Least  Common  Multiple  of  Polynomials. 

9.  Kequired  the  l.  c.  ni»  of  ^^  +  ^  and  a^  —  1^. 

Analysis.  —  Resolving  operation. 

tlie  quantities    into   their  tt^— Z*^  =  («  +  d)  X  («— J) 

prime  factors,  as    in   the  «3+^  =  («  + J)  X  (tt^— ffJ  +  ^) 

margin,  («  +  &)  is  common         (^^  j)  ><  (^_y^  ^  (cii_ah  +  m  = 
to  both,  and  is  their  </.  c.  d.  a         dz    ,      xq       xi        ^      ' 

(A.rt.  139.)    Now  multiply- 
ing these  factors  together,  taking  each  the  greatest  number  of  times 
it  occurs  in  either  of  the  given  quantities,  the  product  a*—a^b  +  ab^—b* 
is  the  Lcm,  required.    (Art.  148.) 

Second  Method. 

^nce  the  g.  c,  d,  contains  all  the  factors  common  to  both  quan- 
tities (Art.  147,  Prin.  2),  it  follows  if  one  of  them  is  divided  by  the 
g,  c.  d.  and  the  quotient  multiplied  by  the  other,  the  product  will 
be  the  I,  c,  tn.    Hence,  the 

EuLE. — Resolve  the  quantities  into  their  prime  factors 
and  multiply  these  factors  together,  taking  each  the  greatest 
number  of  times  it  occurs  in  either  of  the  given  quantities. 
Their  product  is  the  I.  e,  m.  reqtiired. 

Or,  Find  the  greatest  common  divisor  of  the  given  quanti' 
ties^  and  divide  one  of  them  hy  it.  The  quotient,  multiplied 
ly  the  other,  will  he  their  I,  c,  m* 

10.  Find  the  I,  c.  m.  of  2a  —  i  and  4^2  _  i. 

Solution.— The  g,  c.  d,  is  2a— 1.  Now(4a2— i)-^(2a— i)=(2ffi+ 1) ; 
and  (2a  + 1)  X  (2a— i)  =  4a^— I,  Ans. 

Find  the  I,  c,  m,  of  the  following  quantities : 

11.  x^  —  y"^  and  x^  —  2xy  +  i/^. 

Ans.  u?  —  x^y  —  xy"^  +  y\ 

12.  «2_j2  and  a^  —  tfi. 

13.  a?  —  I  and  x^  ■}-  2X  -\-  i. 

14.  2^2  ^  ^a  —  2  and  6a^  —  a  —  i. 

15.  m^  -j-  m  —  2  and  m^  —  i. 

(See  AppS'Uclix,  p.  284.) 


CHAPTER    VIII. 
FRACTIONS. 

150.  A  Fraction  is  one  or  more  of  the  equal  parts  into 
which  a  unit  is  divided. 

151.  Fractions  are  expressed  by  two  quantities  called  the 
numerator  and  denominator,  one  of  which  is  written  below 
the  other,  with  a  short  line  between  them. 

152;  The  Denominator  is  the  quantity  helow  the 
line,  and  shows  into  Jiow  many  equal  parts  the  unit  is 
divided. 

153.  The  Numerator  is  the  quantity  above  the  line, 
and  shows  how  many  parts  are  taken. 

Tlius,  the  expression  -  shows  that  the  quantity  is  divided  into  h 
equal  parts,  and  that  a  of  those  parts  are  taken. 

154.  The  Unit  or  Base  of  a  fi-action  is  the  quantity 
divided  into  equal  parts. 

155.  The  Ter^ns  of  a  fraction  are  the  numerator  and 
denominator, 

156.  An  Integer  is  a  quantity  which  consists  of  one 
or  more  entire  units  only  ;  as  a,  305,  5,  7. 

157.  A  Mixed  Quantity  is  one  which  contains  an 
integer  and  a  fraction. 

Thus,    a  +  -  is  a  mixed  quantity. 

150.  What  is  a  fraction  ?  151.  How  exprensed?  152.  What  does  the  denomina- 
tor show?  153.  The  numerator?  154.  What  is  the  base  of  a  fraction?  155.  Tlia 
terms  of  a  fraction  ?    156.  An  integer  ?    isz*  ^  mixed  quantity  ? 


FKACTIOKS.  71 

158.  Fractions  arise  from  division,  the  numerator  being 
the  dividend  and  the  denominator  the  divisor.     Hence, 

159.  The  Value  of  a  fraction  is  the  quotient  of  the 
numerator  divided  by  the  denominator. 

Thus,  the  value  of  6  thirds  is  6-1-3,  which  is  2  ;  of  — -  is  3m. 


SIGNS    OF    FRACTIONS. 

X60.  JEvery  Fraction  has  the  sign  +  or  — ,  expresse(J 
or  understood,  before  the  dividing  line. 

161.  The  Dividing  Line  has  the  force  of  a  viiicu^ 
luM  or  parenthesis,  and  the  sign  before  it  shows  that  the 
value  of  the  ivhole  fraction  is  to  be  added  or  subtracted. 

162.  Every  Numerator  and  Denominator  is 

pi-^ceded  by  the  sign  +  or  — ,  expressed  or  understood. 
In  this  case,  the  sign  affects  only  the  single  term  to  which 
it  is  prefixed. 

163.  If  the  Sign  before  the  Dividing  Line  is 

changed  from  -f  to  — ,  or  from  —  to  +,  the  value  of  the 
fraction  is  changed  from  +  to  — ,  or  from  —  to  -f . 

^,  hx  —  ex  —  dx  ,  -,    .    X         hx  —  cx  —  dx 

Thus,  a  -\ =  a  +  o  —  c  —  d;  but  a = 

x  X 

a  —  b  +  e  +  d.  • 

164.  If  all  the  Signs  of  the   Numerator  are 

changed,  the  value  of  the  fraction  is  changed  in  a  corre- 
sponding manner. 

Thus, =  +  a  +  6 ;  but 


158.  From  what  do  fractions  arise  ?  159.  What  is  the  value  of  a  fraction  • 
160.  What  is  prefixed  to  the  dividing  line  of  a  fraction  ?  i6r.  What  is  the  force  of 
the  dividing  line  ?  162.  By  what  is  the  numerator  and  denominator  pi'eceded  ? 
How  far  does  the  force  of  this  sign  extend?  163.  If  the  sign  hefore  the  dividing 
line  is  changed,  what  is  the  efi"ect  ?  164.  If  ,all  the  signs  of  the  numerator  are 
changed  ? 


72  FRACTIONS. 

165.  If  all  the  Signs  of  the  Denominator  are 

changed,   the  value  is  also  changed  in  a  corresponding 
manner. 

Thus,  —  =+a;    but    —  =  -a.     Hence, 

+x  —X 

166.  If  any  two  of  these  changes  are  made  at  the  same 
time,  thej  will  balance  each  other,  and  the  value  of  the 
fraction  wiU  not  be  altered, 

_-        ab       —  ab  —ah  ab 

^-^^  -j  =  -ITi   =  - -J- =  -  — 6  =  +  «• 
..  ab       —  ah        ab  —  ab 


PRINCIPLES. 

167.   The  principles  for  the  treatment   of  fractions  in 
A-lgebra  are  the  same  as  those  in  Arithmetic. 

1°.  Multiplying  the  numerator,  or  \  Multiplies   the 
Dividing  the  denomi^iator,        j        fraction. 

,p,  2X2        4        2         .      ,   2  2 

Thus,  -       =  ^  =  -.     And  -       =  -  • 
6  63  t-i-2      3 

2°.  Dividing  the  numerator,  or      )  r^  •  •  7     .-,    ^ 

nr  1^'  1   •      A-L     7         '     ±        \  Divides  the  fraction. 
Multiplying  the  denominator,   )  -^ 

»^      2^2     I      .    ,  2  21 

6  6  6x2       12      6  • 

3°.  Multiplying,  or  dividing  loth  \  Does  not   change  its 

terms  by  the  same  quantity  )  value. 

-«,       2x2      4      2     I      .    ,  2-T-2     I 

Thus, =  -t  =     =  _.     And =  - 

6x2      12      6      3  6-5-2      3 

4°.  Multiplying  and  dividing  a  \  Does  not  change    its 

fraction  by  the  same  quantity  j  value.    • 

-^      2x2-5-2     2 

Thus, =  - . 

6x2-7-2      o 

165.  If  all  the  signs  of  the  denominator  arc  changed  ?     i66.  If  both  are  changed? 
167.  Name  Principle  i.    Principle  2.    Principle  3.    Principle  4. 


BEDUCTION     OF     FRACTIONS.  73 


REDUCTION    OF    FRACTIONS. 

168.  Heduction  of  Fractions  is  changing  their 
terms  without  altering  the  value  of  the  fractions. 

CASE    I. 

169.  To  Reduce  a  Fraction  to  its  Lowest  Terms. 

Def.  —  The  JLoii^est  Terms  of  a  fraction  are  the 
smallest  terms  in  which  its  numerator  and  denominator 
can  be  expressed.     (Art.  122.) 

1.  Reduce  - — =-^  to  its  lowest  terms. 

i$abcx 

Analysis.— By  inspection,  we  perceive  the  operation. 

factors  S,  a,  b,  and  x  are  common  to  both  terms.  5^  (rax aoa^ 

Cancelling  these  common  factors,  the  fraction  l^abcx         3c 

becomes  .     Now  since  both  terms  have  been  divided  by  the 

same  quantity,  the  value  of  the  fraction  is  not  changed.    (Art.  167, 
Prin.  3.)    And  since  these  terms  have  no  common  factor,  it  follows 

that  are  the  lowest  terms  required.     (Art.  122.) 

Note. — It  will  be  observed  that  the  factors  5,  a,  &,  and  x  are  prime  ; 
therefore,  the  product  e^abx  is  the  g,  c.  d.  of  the  numerator  and 
denominator.    (Art.  121.)    Hence,  the 

Rule. — Cancel  all  the  factors  common  to  the  numerator 
and  denominato'* 

Or,  Divide  both  terms  of  the  fraction  by  their  greatest 
common  divisor.     (Art.  167.) 

2.  Reduce f-  to  its  lowest  terms.  Ans.  -• 

i2abc  3 

3.  Reduce to  its  lowest  terms.  Ans.  — 

2,ac  c  • 


168.  What  is  reduction  effractions?    169.  What  are  the  lowest  terms  of  a  frac- 
tion ?    How  reduce  fractions  to  the  lowest  terms  ? 


14:  REBTTCTIOK     OP     FRACTIOKS. 

Reduce  the  following  fractions  to  the  lowest  terms: 


3^y .  lo.     3^—3^^ 

ga^y^  '     2X^y  —  2xyz 

i2a^l)<?  a  -{-he 


^'     4abcd '  '     (a  -{-  be)  X  x 

I'jh^cxy  x^  —  y^ 

^it^cxy  '     x^  —  y/^ 

^AaWc^  ax  —  x^ 

7.     — •  13.     • 

loSabx^y^  cfi  —  a^ 

c^  —  IP'  a  —  I 


c?  4-  2ab  +  6^  c^  —  2a  -\-  \ 

^—y^  ^  +  1 

^*     x^  —  2xy  +  ^'  ^'     a?  -\^  2xy  +  y^' 


CASE    II. 
170.  To  Reduce  a  Fraction  to  a  JVIiole  or  Mixed  Quantity, 

I.  Reduce  to  a  whole  or  mixed  quantity. 

Analysis.— Since  the  value  operation. 

of  a  fraction  is  the  quotient  of  2a  -\-  4b  ■{-  C ,        c 

the  numerator  divided  by  the  2  2' 

denominator,  it  follows  that  per- 
forming the  division  indicated  will  give  the  answer  required.     Now 
2  is  contained  in  2a,  a  times ;  in  46,  2&  times.     Placing  the  remainder 

e  over  the  denominator,  we  have  a  +  26  +  -,  the  mixed  quantity 
required.    Hence,  the 

Rule. — Divide  the  nunurator  hy  the  denominator,  and 
placing  the  remainder  over  the  divisor,  annex  it  to  the 
quotient. 

Note. — This  rule  is  based  upon  the  principle  that  both  terms  are 
divided  by  the  same  quantity.    (Art.  167,  Prin.  3.) 

170.  How  reduce  a  fraction  to  a  whole  or  mixed  quantity?  Note.  Upon  what 
principle  is  this  rule  baeed? 


KEDTJCTIOK     OF     FRACTIO^-S.  75 

Reduce  the  following  to  whole  or  mixed  quantities : 


ax  —  x^ 

X 

3. 

al--^ 

a 

4- 

V^-^ 

1)  ■\-c 

f 

^  +  C2 

a  —  h 

a^  -{-  a^  —  aa^ 


c?  —  ax 
\2^  ■\-  /i^x  —  zy 
^'     h  —  c  ^'  4X 

CASE    III. 
171.  To  Reduce  a  Mixed  Quantity  to  an  Improper  Fraction. 

1.  Reduce  a  -{-  -  to  the  form  of  a  fraction. 

3 

Analysis.— Since  in  i    unit  there  are                   operation. 
three  thirds,  in  a  units  there  must  be«  b 3^^        b 

times  3  thirds,  or  3^;  and  3?  +  -^=^^,  ""        3~7        3 

3  3       3         3  3a        h       3a-{-b 

the  fraction  required.    Hence,  the  1 —  = — 

3        3  3 

Rule. — Midtiply  the  integer  hy  the  denominator  ;  to  the 
product  add  the  numerator,  and  place  the  sum  over  the 
denominator.     (Art.  65.) 

Notes. — i.  An  integer  may  be  reduced  to  the  form  of  a  fraction  by- 
making  I  its  denominator.     Thus,  a  =  - . 

2.  If  the  sign  before  the  dividing  line  is  —  and  the  denominator  is 
removed,  all  the  signs  of  the  numerator  must  be  changed.  (Arts.  163, 82.) 

Reduce  the  following  to  improper  fractions  : 

,       cd  .        ahd  —  cd         , 

2.  at) i-  Ans,  3 =  ao  ^  c, 

d  d 

xy  —h  ,  .   I  —  ^ 

3.  3X  +  -^— 6.     a;  —  I  + 


\  •\-  X 
T  ^   a  —  c  a  —  t 

S.     a  +  l  +  ^y  S.     8.  +  3^. 


OPEBATIOH. 

3«  = 

3« 
I 

I 

X  5  _ 

X5 

isa 
5 

Hence,  the 

76  EEDUCTION     OF     FKACTI0N8. 


CASE     IV. 

172.  To  Reduce  an  Integer  to  a  Fraction  having  any  required 

Denominator. 

1.  Reduce  3«  to  fifths. 

Analysis. — Since  in  i«  there  are  5  fifths,  in 
3a  there  must  be  3  times  5  fifths,  or  -^. 

Or,  reducing  the  integer  3a  to  the  form  of  a 
fraction,  it    becomes    — ;     multiplying  both 

terms  by  the  required  denominator,  we  have  -^, 

Rule. — Multiply  the  integer  hy  the  required  denominator, 
and  place  the  product  over  it. 

2.  Reduce  2x  to  a  fraction  having  6m  for  its  denominator. 

3.  Reduce  6ax  to  a  fraction  having  4ab  for  its  denominator. 

4.  Reduce  ^a  +  4b  to  a  fraction  having  6c^  for  its 
denominator. 

5.  Reduce  x  —  y  to  a  fraction  having  x  -i-  y  for  its 
denominator. 

6.  Reduce  2x^y  to  a  fraction  having  ^a^  —  2b  for  its 
denominator. 

173.  To  Reduce  a  Fraction  to  any  Required  Denominator. 

I.  Change  -  to  a  fraction  whose  denominator  is  12. 

Analysis. — Dividing  12,  the  required  de-  ofebatiok. 

nominator,  by  the  given  denominator  3,  the  ^  2  -^  3  =  4 

quotient  is  4.     Multiplying  both  terms  of  a  X  4 4« 

the  given  fraction  by  the  quotient  4,  the  *  2X4        1 2 ' 

result,  — ,  is  the  fraction  required.    Hence,  the 

Rule. — Divide  the  required  denominator  hy  the  denomi- 
nator of  the  given  fraction,  and  multiply  loth  terms  hy  the 
quotient. 

172.  How  reduce  an  integer  to  a  fraction  having  any  required  denominator? 


REDUCTION     OF     FRACTION'S.  77 

Reduce  the  following  to  the  required  denominators : 

2.  Reduce  —  to  thirty-fifths. 

Solution.    35  -5-  7  =  5.    ^o\^ =  — .    Ans. 

7-^5        35 

3.  Reduce  -  to  the  denominator  ac, 

c 

4.  Reduce  —  to  the  denominator  49a. 

7 

5.  Reduce  ^  to  the  denominator  a?  —  2xy  -f  ^. 

X  —  y 

6.  Reduce  -— —  to  the  denominator  M^(x  +  ti)\ 

x  +  y  \        JJ 


COMMON    DENOMINATORS. 

174.  A  Common  Denominator  is  one  that  belongs 
equally  to  two  or  more  fractions. 

PRINCIPLES. 

1°.  A  common  denominator  is  a  multiple  of  each  of  the 
denominators  ;  for  every  quantity  is  a  divisor  of  itself  and 
of  every  multiple  of  itself     (Art.  124,  Prin.  i.)     Hence, 

2°.  The  least  common  denominator  is  the  least  common 
multiple  of  all  the  denominators, 

CASE    V. 

175.  To  Reduce  Fractions  to   Equivalent  Fractions  having  a 

Coinmon  IJenoitiinaior. 

fh  C  X 

I.  Reduce  t>  ;t>  and  -,  to  equivalent  fractions  having 
a  common  denominator. 


173.  How  reduce  a  fraction  to  any  required  denominator  ?    174.  What  is  a  com- 
jnon  denominator  ?    Principle  i  ?    Principle  a  ? 


78  REDUCTION     OF     FRACTIONS. 

Solution.— Multiplying  the  denominators  h,  d,  and  y  together,  we 
have  bdy,  which  is  a  common  denominator. 

b  X  d  X  i/  =  hdy       The  common  denominator. 

a  X  d  X  ^  =  ady  ) 

ex   i   X  y  =  hey    V  The  new  numerators. 

X  X  l  X  d  =z  hdx  ) 

J  a ady         c hey         x hdx 

h~  hdy'        d~~hdy'        y^  hdy' 

To  reduce  the  given  fractions  to  this  denominator,  we  multiply- 
each  numerator  by  all  the  denominators  except  its  own,  and  place  the 
results  over  the  common  denominator.    Hence,  the 

Rule, — Multiply  all  the  denominators  together  for  a  com- 
mon denominator,  and  each  numerator  into  all  the  denom- 
inators, except  its  oion.for  the  neio  numerators. 

Notes. — i.  It  is  advisable  to  reduce  the  fractions  to  their  loicest 
terms,  before  the  rule  is  applied.     (Art.  169.) 

2.  This  rule  is  based  on  the  principle,  that  the  Dolue  of  a  fraction  is 
not  changed  by  multiplying  both  its  terms  by  the  same  quantity. 
(Art.  167,  Prin.  3.) 

Reduce  the  following  to  equivalent  fractions  having  a 
common  denominator: 


2. 

a     X     ^ 
~c'  y'  4* 

Ans.  4^^    4^^    ''^-. 
4cy      4cy     4cy 

3. 

c      h     2d 
d'  x'  T 

8. 

2a      c  -\-  I 

4. 

a       h      X 
2x'   ~?'   ~y 

9. 

2      a      c  -\-  d 
3'    ^'   c-d 

5. 

2a         X 

10. 

xy     I      2« 

Sh'    a  +  b 

z'    2'     h' 

6. 

x  —  y     x^y^ 
x-\-y'    x  —  y' 

II. 

2a             x^  +  ^^ 

7. 

a  +  h     5a—  1 
3     '        a      ' 

12. 

a  —  x     a  +  X 

a  -{-  x'    a  —  x 

175.  How  reduce  fractions  to  equivalent  fractions  having  a  common  denomina- 
tor r       JV^oie.  Upon  what  principle  is  this  rule  based  ? 


EEDUCTIOK     OF     FRACTIONS. 


79 


CASE    VI. 
176.  To  Reduce  Fractions  to  the  Least  Common  Denominator. 

I.  Eeduce  - 

X'  xy  yz 

Solution. — Tlie  I,  c,  m,  of  the  denominators  U 


,  and  —  to  the  I,  c,d. 


xyz  =  L  c,  d. 


xyz  -r-x   =  yz 
xyz  -T-xy  =i  z 

xyz  -i-  yz  ^x 


The 
multipliers. 


a  X  yz 
X  X  yz 
b  X  z 
xy  X  z 
d  X  X 


xyz.     (Art.    >j8. 

_  ^?/^ 
~xyz 
_bz^ 
~  xyz 
dx 


The 
fractions 
required. 


yz  X  X      xyz 

To  change  the  given  fractions  to  others  whose  denominator  is  xyz,  we 
multiply  each  numerator  by  the  quotient  arising  from  dividing  this 
multiple  by  its  corresponding  denominator.    Hence,  the 

Rule. — I.  Find  the  least  common  multiple  of  all  the 
denominators  for  the  least  common  denominator. 

II.  Divide  this  multiple  by  the  denominator  of  each  frac- 
tion, and  multiply  its  numerator  iy  the  quotient. 

Note. — AH  the  fractions  must  be  reduced  to  their  lowest  terms 
before  the  rule  is  applied. 

Reduce  the  folloAving  fractions  to  the  I,  c.  d, : 


a      Ic      y 

2. 

2^'    a: '    4C 

cd     2X     xy 

3- 

ab^    2,a^   ac 

a     i     c      X 

4- 

~ '    'Z'    T '    T.' 

2    3    4    y 

a^c     2cd     x^y 

S- 

ah'    Wc'    ^U 

6. 

2db     3       a;       I 

Zac'    4'    OL^c'    8 

2«     cd     x^y 

h 

45 '    Ic'    lex 

8. 


10. 


II.     — 


12.     -^ 


a^l  a  — I  a^+ly^ 
a  —  y  a  +  V  a^-lj'' 
2(x-\-y)     a  ah 

zix-^tjY  xy'    'ejx'-^y)' 
d       X 
a^'    Wb' 
X       m       y 
ac    Wc'   &d 
X      a  ■\-h      d 
'   xz 


y 


2' 


xy 


13- 


m-\-n     m—n 


m* 


W 


2ax^  '   ^cx 


176.  How  reduce  fractions  to  the  least  common  denominator! 


so  ADDITION     OF     FRACTIOKS 


ADDITION    OF    FRACTIONS. 

177.  When  fractions  have  a  common  denominator,  their 
numerators  express  like  parts  of  the  same  unit  or  base,  and 
are  like  quantities,     (Art.  43.) 

178.  To  Add  Fractions  which  have  a  Common  DenominatoPc 

1.  What  is  the  sum  of  f ,  |,  and  |  ? 

Solution,    f  and  |  are  V-,  and  f  are  ^i,  Ans.    Hence,  the 

EuLE.— ^flf^  the  numerators,  and  place  the  sum  over  the 
common  denominator, 

.  ,,  75      45,85  .        loj 

2.  Add  —  ,   —  ,  and  — .  Ans,  -^— 

m      m  m  m 

3.  Add  ^ — , , ,  and  ^ 

2xy      2xy      2xy  2xy 

.  ,,  7c?a:2!      i7(?a;2!      iidxz         ,  4<7a;;? 

4.  Add  '—r ,         ,     ,    — 7-  ,  and  ^^  • 

$aoc      ^aoc       $aoc  ^aoc 

5.  Add  ?^+f  to  3i-rf .         6.  Add  ^^±^  to  4^=^. 

179.  To  Add  Fractions  which  have  Different  Denominators. 

7.  What  is  the  sum  of  t^   -7>  and  —  ? 
h     d'  X 


bxdxx  =  hdx,  c. d, 
a      adx        c      bcx 


Analysis, — Since  these  fractions 

have    different   denominators,    their  

numerators  cannot  be  added  in  their  5        odx         d       udx 

present  form.    (Art,  66.)    We  there-  m        hdm 

fore  reduce  them  to  a  common  denomr  ^        ~^x 

inator,    then    add    the    numerators.  ^^^  _^  j^^.  ^  j^^ 

(Art.  175.)    Hence,  the  j^ -,  Ans^ 

Rule. — Reduce  the  fractions  to  a  common  denominator, 
and  place  the  sum  of  the  nuriierators  over  it. 

Note.— All  answers  should  be  reduced  to  the  lowest  terms. 


177.  When  fractions  have  a  common  denominator,  what  is  true  of  the  numerators! 
178.  How  add  such  fi:3,ction8  ?    179.  How,  whea  they  have  different  denominators  ? 


ADDITIOI^     OF     FRACTIONS.  81 


Find  the  sum  of  the  following  fractions : 


8.     --f--^  + 
V        X        m 


2X       2y       2xy  .        2X^m  -\-  lyhn  -f  2X^y^ 

mxy 
cd      Ay       Ix 

^        ZX^  2d^    S 


9- 

3^  +  ^  +  1 

4        5       3 

lO. 

^  +  J-  +  M. 

3        26?        4 

a      .       X 

II. 

I  ■\-c  '  l  —  c 

12. 

x+y      x—y 

2xy           xy 

I-?- 

2  +x      3 -{-ax 

y          ay 

^  A. 

a             ah 

i6. 


a       271  -\-  d 


d  sh 

a          d 
17.     -  H 

y       -m 

,8.    :^+    -^ 


y         m  —  n 

—  4       —  16 

19.  — ^  + 

2         7  —  3 

4«       6c       xm 

20.  -^^ — — • 

X  -^  y      x  —  y  b        d       2>x 

180.  To  Add  Mixed  Quantities. 

Ji  vn 

1.  What  is  the  sum  of  «  +  -  and  d ? 

c  n 

BOLunoiS'. — Adding  the  integral  and  fractional  parts  separately^ 

\06  result  S&a^r  d  ^ ,  tlie  sum  required.     Hence,  the 

en 

Efle. — Add  the  integral  and  fractional  parts  separately, 
and  unite  the  results.     (Art.  179.) 

Note. — Mixed  quantities  may  be  reduced  to  improper  fractions,  and 
tlien  be  added  by  the  rule.    (Art.  171.) 

2.  What  is  the  sum  of  «  +  -  and  c  +  -  ? 

2  X 

3.  What  is  the  sum  oi  x  -\-  ^  and  ? 

0  m  —  y 

4.  What  is  the  sum  of  3d and  a  -\ ? 

2  I 

a  ~—  1/ 

5.  What  is  the  sum  of  5a;  +  ^  and  — ^  ? 

0  2 

180.  How  add  mixed  quantities  ?    iVofe.  How  else  may  they  be  added  ? 


82  SUBTRACTION     OF     FRACTIONS. 

181.  To  Incorporate  an  Integer  with  a  Fraction. 

c ^ 

6.  Incorporate  the  integer  ah  with  the  fraction • 

Solution. — Reducing  db  to  the  denominator  of  tlie  fraction,  we 

have  ah  =  '^^^^M  .  and  3_«&^^  +  -£z:i  =  'i^x^^y^;C-d  ^^^ 

2,x+y  2>^+y         o^x+y  y^^y 

Hence,  the 

EuLE. — Reduce  the  integer  to  the  denominator  of  the  frac- 
tion, and  place  the  sum  of  the  numerators  over  the  given 
denominator,     (Art.  172.) 

za 

7.  Incorporate  the  integer  3^  with  the  fraction  -r- 

8.  Incorporate  —  4^  with  « 

9.  Incorporate  —  a  with         ^« 

a  —  "b 


10.  Incorporate  z^  +  V  with 


x  —  y 


II.  Incorporate  —  a  -\-  ^h  with ^• 


12.  Incorporate  2a;  +  2y  with 


iC 


SUBTRACTION    OF    FRACTIONS. 

182.  The  numerators  of  fractions  which  have  a  common 
denominator,  we  have  seen,  are  like  quantities,  (Art.  177.) 
Hence,  they  may  be  subtracted  one  from  another  as  integers. 

1.  Subtract  |  from  |. 

Solution.     l-l^T^^^l  or -.    Am,  - • 
00004  4 

2.  From  T  subtract  y  ^ns.  t  —  7  =  — 7 — • 

6  J  III 

181.  How  incorporate  an  integer  with  a  fraction?  182.  What  is  true  of  the 
numerators  of  fractions  having  a  common  denominator  ?  How  suhtract  euch  frac- 
tions ? 


3 

X   2  = 

6,  c. 

.c?. 

7«_ 
3 

14a 
6 

2 

ga 
6 

I4« 
6 

9«_ 
6  " 

6' 

An 

SUBTRACTION     OF     FRACTIONS.-  83 

3.  From  —7—  subtract  -^-^ 

4.  From  — -^  subtract  - — • 
^  a  a 

183.       To  Subtract  Fractions  which  have  Different 
Denominators. 

q.  From  --  subtract  —  • 
^  3  2 

Analysis. — Since  these  fractions 
have  different  denominators,  they  can- 
not be  subtracted  one  from  the  other 
in  their  present  form.  We  therefore 
reduce  them  to  a  common  denominator , 
which  is  6,  and  place  the  difference  of 
the  numerators  over  it.    Hence,  the 

RcTLE. — Reduce  the  fractions  to  a  common  denominator, 
and  subtract  the  numerator  of  the  subtrahend  from  that 
of  the  minuend,  placing  the  difference  over  the  common 
denominator. 

Notes. — i.  The  integral  and  fractional  parts  of  mixed  quantities 
should  be  subtracted  separately,  and  the  results  be  united. 

Or,  mixed  quantities  may  be  reduced  to  improper  fractions,  and  then 
be  subtracted  by  the  rule.     (Art.  171.) 

2.  A  fraction  may  be  subtracted  from  an  integer,  or  an  integer 
from  a  fraction,  by  reducing  the  integer  to  the  given  denominator, 
and  then  applying  the  rule. 

6.  From  5«-^  take^^^.  Ans,  5^:iJ£^. 

X  '  y  xy 

7.  From  — ,  take  • 

'  m'  y 

8.  From     ~~    ,  take • 

m  y 

^         a  -\-  -Kd    ^  ^      m —  2d 

9.  From  — '-^j  take  ^ 

4 3 

.•83.  How  when  they  have  different  denominators  ?  Note  i.  How  subtract  mixed 
quantities  ?    Note  2.  How  a  fraction  from  an  integer,  or  an  integer  from  a  fraction  T 


84  MULTIPLICATION^     OF     FEACTIONS, 

10.  From  — ,  take  —  -         » 

m  y 

11.  From  -,  take  m, 

y 

12.  From  4«  +  -,  take  3a  —  :^' 

c  d 


2                 3 

14.  From  T-^—,  take   ,  ^    ■. 

15.  From  «  — -,  take  ^. 
f            2 

16.  From  "^  -  ^  take  "^-^ 
10   '         i^  +  y 

n.  From  a;      ^""'',  take  ^""^ 

-a. 

MULTIPLICATION    OF    FRACTIONS. 
CASE    I. 
J84.  To  Multiply  a  Fraction  by  an  Integer, 

1.  Multiply  -7  by  m. 

A-{^ALysis.— Multiplying  the  numerator  of  the  ofkiutioi. 

fhftCtiun    by  the   integer,  the  product   is   am.  ^  ^  ^ ^^^ 

(Art.  i6;,  Piin.  i.)  b  ~  b   ' 

2.  "W^at  :s  the  product  of  7-  x  a;? 

Analysis- 'A  fraction  is  multiplied  by  ^  y^  x ^ 

dividing  its  denominator  ;   therefore,  if  we  bx  bx  -r-  X 

divide  hx  by  gc,  the  re.sult  will  be  the  product  a  a 

required.    (Art.  167,  Knn.  1.)    Hence,  the  ^^"ZZTJ  ^^  5* 

EuLE. — Multivly  i?a  numerator  by  the  integer. 
Or,  Divide  the  detwminaior  by  it. 

Notes. — i.  A  fi'action  is  multiplied  by  a  quantity  equal  to  its 
denominator,  by  cancelling  the  denominator.     (Art.  1 10,  Prin.  4.) 

184.  How  multiply  fi  frdctioo  by  on  integer? 


MULTIPLIOATIOK     OF     FRACTIONS.  85 

2.  A  fraction  is  also  multiplied  by  dJij  factor  in  its  denominator,  by 
cancelling  that  factor.    (Art.  no,  Prin.  4.) 


Find  the  product  of  the  following  quantities: 
3- 


4 

5 
6 

7. 
8. 

9 

10 

II 


^2/ 

— — T  X  (a  — -  J).  -4?^5.  3a;, 


—y     X     d. 

cd 

—7-^  X  (3  +  m). 

«^ 

—  X  6. 

24 

2a;  —  XII  ,  _ 

v^~i  X  (3c  +  2fZ).     14. 

ax  —  X  6x. 

SX 

a  +  h 

; X  ^x. 

20X  4-  2^xy      ^ 

a  -\-  ab 

he  -\-  c 

2X-\-  s 


X  2ac. 


X  2or2;. 
5  a^  —  z^ 


4       «^ 
Ans,  —  • 
c 

12. 

a  —  i       , 

^      X  (12a:  +  18). 

13- 

-J X  (a  —  x), 

d  —  x       ^          ' 

14. 

a  +  b 

15. 

40Z  —  10       ^            ^ 

16. 

3c  —  d 
20      >^'5. 

17. 
18. 

-;^ X    X    12;  +  «). 

CASE    II. 
185.  To  Multiply  a  Fraction  by  a  Fraction^ 

I.  What  is  the  product  of  -  by  — 

Analysis. — Multiplying   the   numerator  of  operation. 

the    fraction    -  by  d,  the  numerator  of   the  axd_ad 

^  ad  C  c 

multiplier,  we  have  — ,    But  the  multiplier  is  -j  -, 

— ;  hence  the  product  —  is  JW  times  too  lar^e.  C    X  ffl       cm 

^  d  a 

To  correct  this,  multiply  the  denominator  —  X  —-=:-- 

by  m.    (Art.  167,  Prin.  2.)  ^        ^-        ^^ 

Note  I.  How  is  a  fraction  multiplied  by  a  quantity  equal  to  its  denominator  J 
Note  2.  How  by  any  factor  in  its  denominator  ? 


e6  MULTIPLICATION     OF     FRACTIOITS. 

2.  Required  the  product  of  — -^  multiplied  by  — • 

Analysis. — The  factors  2,  r,  and  e  are  opebation. 

common  to  each  term  of  the  given  frac-  2al)        cm 2abcm 

tions.    Cancelling  these  common  factors,  6cd        ax       6acdx 

the  result  is  -^  ,  the  product  required  2aocm om      . 

ddcdj'jc        1(1  cr 
(Art.  167,  Prin.  3.)    Hence,  the  ^ 

Rule. — Cancel  tlie  common  factors;  then  multiply  the 
numerators  together  for  the  new  numerator,  and  the  denom- 
inators for  the  new  denominator. 

Notes. — i.  Mixed  quantities  should  be  reduced  to  improper  fractions, 
and  then  be  multiplied  as  above. 

Or,  i^Q  fractional  and  integral  parts  may  be  multiplied  separately, 
And  the  results  be  united. 

2.  Cancelling  the  common  factors  shortens  the  operation,  and  gives 
the  answer  in  the  lowest  terms. 

3.  The  word  of  in  compound  fractions  has  the  force  of  the  sign  x . 
Therefore,  reducing  compound  fractions  to  simple  ones  is  the  same  as 
multiplying  the  fractions  together.    Thus,  |off  =  fxf  =  ^. 


Find  the  products  of  the  following  fractions: 

2X      xxii      2dy  ^13 

3.  —  X  — I  X  — ^«  6.    — ; X  I- 

^      y        2d        X  a  +  3a:      8 

he      X       d  (a  +  m)y.h  aii 

4.  —  X  T-  X  —  7.     ^^ —  X  -r-~--{ — 

a       by      c  IX  {a-{-m)xc 

x-^y  ^x  +  y  a±h      cd 

5»      X   ; — •  o.      5 —  X  — • 

yz        y  +  z  (?         X 

9.  What  is  the  product  of  -^—  by  — ^^^  ? 

Solution. — Factor  and  cancel.    Ans.  —(x+y). 

185.  How  multiply  a  fraction  by  a  fraction  ?  Note  i.  How  mixed  quantities ! 
Vote  2.  How  shorten  the  operation  ?  Note  3.  What  is  the  force  of  the  word  0/  ia 
compound  fractiouB  ? 


MULTIPLICATION     OF     FEACTI0:N^S.  87 

HT  li.-  1        2a     ,     x^  —  y^  .        2a  (x  +  y) 

10.  Multiply by  ^-  Ans.  — —j-^' 

11.  Multiply  -^  by    \~       * 

^  "^       4a:        "*  y^  —  2xy 

TIT  1  J.-  1     4^  —  2h   ,      2a  —  h 
.2.  Multiply  ^—--^  by  -^. 

13.  Multiply  «  +  -^-  by  ^. 

14.  Multiply  ^  +  1^  by  -t/. 

15.  Multiply  a;  —  —  by  -  +  ^. 

16.  Multiply  «  4-  -T-  by  — y 


CASE    III. 

186.  To  Multiply  an  Integer  by  a  Fraction* 

dx 
I.  Multiply  the  integer  a  by  — 


Analysis. — Changing   the   integer  to  the 

a  «  = 


a 


form  of  a  fraction,  we  have  -  to  be  multiplied  I 

.     dx      ...  .    adx^  ^        ^.  a      dx  _adx 

by  — ,  which  equals  .    Hence,  the  7  ^  77;  —  "777  * 

^  cy*  ^         cy  1       cy       cy 

Rule. — Reduce  the  integer  to  a  fraction  ;  then  multiply 
the  numerators  'together  for  the  new  numerator,  and  the 
denominators  for  the  new  denom,inator. 

Notes. — i.  Multiplying  an  integer  by  b.  fraction  is  the  same  as  find- 
hig  B.  fractional  part  of  a  quantity.     Thus,  a;  x  |  is  the  same  as  finding 

f  of  X,  each  being  equal  to  — .    That  is, 
4 
2.  Three  times  i  fourth  of  a  quantity  is  the  same  as  i  fourth  of 
3  times  that  quantity. 

186.  How  multiply  an  integer  by  a  fraction  ?  Note.  To  what  is  this  operation 
similar? 


88  MULTIPLICATION     OF     FRACTION'S. 

Find  the  product  of  the  following  quantities  : 

9.     {x'-f)x        ""' 


2. 

,        dx 

abc  X  — 

cy 

3. 

,       b  -\-c 
ad  X  — ' — 
xy 

4. 

ax  X  ' 

4« 

5. 

(«  +  ^)xf 

3  (^  +  «/) 


^  2  («  +  /^) 

II.     (:z:2  +  i)  X        ^^^ 


3(^—0 
12.     2a;^(flj  —  Z*)  X 


;2  — Z»2 


6-     (3«  —  2/)  X  ^-  13.     3a (2;  —  i)  X  ~^^ 


y  *.      ^    V  '      ^'^  _  I 

7.  (-^+0X^.  14.       (2..  +  .2)x^^. 

8.  (i~«^)x-^.         15.     (I-^^2)x      ' 


I  + «  ^  '       n-\-  \ 

187.  The  principles  developed  in  the  preceding  cases  may 
be  summed  up  in  the  following 

GENERAL    RULE. 

Reduce  whole  and  mixed  quantities  to  improper  fractions , 
then  cancel  the  common  factors,  and  place  the  product  of  the 
numerators  over  the  product  of  the  denominators, 

1.  Multiply by  15a;. 

2.  Multiply     ^^     by  y^  —  i. 

3.  Multiply  ^  +  —  by  —^. 

,,  ,,.  ,     2ax      ^ab  ,      xac 

4.  Multiply  —  X  — -  by  --^. 

^  "^     a         ac     ''  2ab 

5.  Multiply  ^-  by  ^. 
^  ^ -^   loy    ^  ga 

6.  Multiply  X  7  by 


a  a  -\-  b    ''  a 


187.  What  Ib  the  general  rule  for  multiplying  fractions  ? 


DIVISION     OF     FRACTIONS.  85 


7.  Multiply  ^— —  by  x  +  z. 

8.  Multiply  ^  by  2y\ 

if 

9.  Multiply by  a;^  —  2xy  +  «/2. 

a;  —  y 

10.  Multiply  - — —-^  by  8;?  —  2. 

^  -^  402;  —  10     -^ 

11.  Multiply  a;  —  -^  by  -  +  ^» 

^  -^  X     -^  y      X 

2a^        2ah 

12.  Multiply  a  +  ^  by  — ^  • 

.,  Multiply  (i±l)ib,^^. 

14.  Multiply  ^  by  -5 ^' 

^  -^       4a;  y^  —  2xy 

15.  Multiply  &  +  ^^^  by  Z»  -  ^^^. 

2//  ^2  ^2 

16.  Multiply  by —- 

^  -^  x  —  y    *'       ax 


DIVISION    OF    FRACTIONS. 

CASE    I. 
188.  To  Divide  a  Fraction  by  an  Integer, 

I.  If  3  oranges  cost  —  dollars,  what  will  i  cost? 

71/ 
Analysis.— One  is  i  third  of  3;  therefore,  opeeation. 

oa  9^   .       _  3^ 

I  orange  will   cost  i   third  of  —  dollars,  and  "^  "^  3  —  ~ 

-  of  —  dollars  is  —  dollars.    (Art.  167,  Prin.  2.) 


2.  Divide  -  by  m. 

c     ^ 

Analysis. — Since  we  cannot  divide 


OPBEATION. 


the  numerator  of  the  fraction  by  m,         —  -i-  m  rz=  ^ z=.  — • 

we  multiply  the  denominator  by  it.         C    '  C  X  m        cm 


90  DIVISION     OP     FRACTIONS. 


The  result  is  — .     For,  in  eacli  of  the  fractions  -  and  — ,  the  same 
cm  c  cm 

number  of  parts  is  taken ;  but.  since  the  unit  is  divided  into  m  times 

as  many  parts  in  the  loiter  as  in  the  former,  it  follows  that  each  part 

in  the  latter  is  only  — th  of  each  part  in  the  former.     Hence,  the 
''   m 

Rule. — Divide  the  numerator  by  the  integer. 
Or,  Multiply  the  denomhiator  hy  it. 

Note. — If  the  dividend  is  a  mixed  quantity,  it  should  be  reduced 
to  an  improper  fraction  before  the  rule  is  applied.    (Ex.  3.) 


Divide  the  following  quantities 

3. 

a  -\ by  d,^ 

X 

4. 

^  +  l^hjxy. 

5. 

/  by  s^y- 

9- 

6. 

2«^    1         X 

—  by  5. 

sac    -^ 

10. 

7. 

«  H hy  a, 

c 

II. 

8. 

ax  H by  xK 

12. 

.       ax  -\-  be 

Ans.  :: 

dx 

.       abx  4-  11 
Ans,  — J — -• 
abxy 

c^  -\-  ax  , 

-^ —  by  a-\-  X, 

2b         '' 

a^  —  <^  , 

^— - —  by  a  —  c. 

7?  +  2xy  +  if  . 

— — — ^—^-^~  by  x-\-y. 

— ■ — f  by  «  +  J. 
a  —  b^ 


CASE    II. 
189.  To  Divide  a  Fraction  by  a  Fraction, 

This  case  embraces  two  classes  of  examples : 

First.    Those  in  which  the  fractions  have  a  common 
denominator. 

Second.  Those  in  which  they  have  different  denominators. 


188.  How  divide  a  fraction  by  an  integer?    NoU.  If  the  dividend  is  a  mixed 
quantity,  how  proceed  ? 


DIVISION^     OF     FRACTIONS.  91 

1.  At  —  dollars  apiece,  how  many  kites  can  a  lad  buy  for 

dollars  ? 

m 

Analysis.— Since  these  fractions  have  a  com-  opbbation. 

mon    denominator,   tlieir  numerators    are  like  12a    ^    ^a 

quantities,   and   one   may  be  divided    by  the  ni     '    m 

other,  as  integers.     (Art.  177.)     Now  yi  is  con-  Ans,  4  kites. 

tained  in  12a,  4  times.    (Art.  no,  Prin.  i.) 

2.  How  many  times  is  - —  contained  in  -^—  ? 

''  X  X 

3.  What  is  the  quotient  of  ^^  divided  by  — ^- 

n        c 

4.  It  is  required  to  divide  -  by  -• 

X       y 

Analysis. — Since  these  frac- 
tions have  different  denomina- 
tors, their  numerators  are  unlike 
quantities ;  consequently,  one 
cannot  be  divided  by  the  other 
in  this  form.  (Art.  114,  no^e.)  We 
therefore  reduce  them  to  a  com- 
mon denominator ;  then  dividing 
one  numerator  by  the  other,  the 
result  is  the  quotient. 

Or,  more  briefly,  if  we  invert  the  divisor,  and  multiply  the  dividend 
by  it,  we  have  the  8am£  combinations  and  the  same  result  as  before. 
(Art.  185.)    Hence,  the 

EuLE. — Multiply  the  dividend  hy  the  divisor  inverted. 
Or,  Reduee  the  fractions  to  a  eommon  denominator,  and 
divide  the  numerator  of  the  dividend  hy  that  of  the  divisor. 

Notes. — i.  A  fraction  is  inverted,  when  its  terms  are  made  to 

change  places.    Thus,  ^  inverted,  becomes  -  • 
0  a 

2.  The  object  of  inverting  the  divisor  is  convenience  in  multiplying. 

3.  After  the  divisor  is  inverted,  the  common  factors  should  be  can^ 
celled  before  the  multiplication  is  performed. 

189.  How  divide  a  fraction  by  a  fraction  when  they  have  a  common  denominator  ? 
When  the  denominators  are  different,  how  ? 


rmST  OPERATION. 

a  _ 
x~ 

_ay              c  _cx 
~  xy'           y~~  xy' 

ay  ^  ex  _ay 
xy  '  xy~  ex 

SECOKD  OPEBATIOK. 

a 

X 

y      X      c       ex 

92  DIVISION     OF     FBACTIONS. 

Divide  the  following  fractions : 

^      cd  ^  ay  ^2*      2¥c^    ^  6bc 

6.  ^-r-  by  -^.  14.     —  by  ^— . 

taoy    ''ax  4cd      ''    2d 

7.  3^  by  a  ,5.     J^by^f"?. 

8.  -^Lby^  t6.     A^by3^^-?. 
a;— 1*^2  a  -\-  h    ''     2y 

g—  I  1^    _^  a:*  ,     «a? 

,0.     5^  by -'-5^.  ,8.     36«f!i,    £8aS_ 

„.     iil£±^)  byii^±^.      10.     -SLbyl^. 

12.     -^-r-i  by 20.     -—4  by  -f— ^. 

a  +  0    "^     a  iSab    *^  36^6? 

CASE    III. 
190.  To  Divide  an  Integer  by  a  Fraction. 

I.  Divide  the  integer  ydc  by  ^' 

Analysis.— Having  reduced  the  opbbation. 

integer  to  a  fraction,  and  inverted  ndc  -^  ^       = 

the  divisor,  we  cancel  the  common  ^ 

factor  d,  and  proceed  as  in  the  last  jdc  h  ']hG       . 

case.    Hence,  the  ~i     ^  ^  ~  3«  '  ' 

Rule. — Reduce  the  integer  to  a  fraction,  and  multiply  it 
hy  the  divisor  inverted. 

Divide  the  following  quantities : 

^  '    ■      jx 

190,  How  divide  au  integer  by  a  fraction  ? 


2. 

,         ex 
'^y  ■■  dm 

3- 

ax:      ^    . 
m  -{-  n 

4. 

J/ 

DIVISION     OF     FRACTION'S. 


93 


191.  Complex  JFractions  are  reduced  to  simple  ones, 
by  performing  the  division  indicated. 


8.  Keduce  —  to  a  simple  fraction. 
3 


Analysis. — The  given  fraction  is  equiv-  a 


alent  to 


«  .  3 


.     Performing  the   division 


b 


,  the  dividend. 


indicated,  we  have  ^,  the  simple  fraction 
required.    (Art.  189.) 


-,  the  divisor. 
4 

^  y  4  _  4a     . 

b       3       3b 


Reduce  the  following  fractions  to  simple  ones : 


10. 


II. 


b 
Id" 
«4-i 

«  —  I 

a—  I 

a+  I 
a-b 

^  +  .V 

x  —  y 

a  +  b 

Ans. 


12. 


13- 


bed 


a?  —  y'^ 

a-b 

x  +  y 
x-\-y 

a-^b 

a  +  b 

x  —  y 

192.  The  various  principles  developed  in  the  preceding 
cases  may  be  summed  up  in  the  following 


GEN  ERAL     RULE. 

Reduce  integers  and  mixed  quantities  to  improper  frac- 
tions, and  complex  fractions  to  simple  ones  :  then  multiply 
the  dividend  by  the  divisor  inverted. 


191.  How  reduce  complex  fractions  ?    192.  What  is  the  general  rule  for  dividing 
fractionB? 


94 


DIVISION     OF     FKACTIONS, 


1.  Divide  — —  by  sic 

4xyz 

2.  Divide  ^^  ^     by  gxy. 

2'jy 

^.  .,     i6xy  ,      2cd 
-?.  Divide by  — 

^  fyn.       ''     inn. 


^.  .,       42ah     ,        145 

4.  Divide  -^ 5  by 

5.  Divide  -^^^-^  by  -^. 

,    TA.  .,    x'^  —  2xy  +  ifi  ,     ^  —  y 

6.  Divide  /       -^    by  ^. 

7.  Divide  -X ; — -„  by 

Solution. — Factoring  and  cancelling,  we  have, 

a^—m*       _  (a®  +  m^) (a  +  m){a—m)  ^   a? -{■am  _  a(a  +  m)     ^ 
a^—2am  +  m^~         {a—m)(a—m)        '     a—m  ~    a—m 

(a^-\-m-)(a  +  m)(a—m)        a—m        a'^  +  m^  m^     . 

' ^-^, — -\ X  -7 ^  = ,    or    a  +  — »  Ans, 

(a—m)  {a—m)  a{a  +  m)  a  a 

8.  Divide  -^^  by  -^—. 

cfi  —  x^    ''a  —  X 

-p. .  . ,     4^2  —  8c  ,      c2  —  4 

0.  Divide by -* 

^  x  +  y      ''  X  -\-  y 

10.  Divide  ; by  ^-73-- — (• 

ac  -^  ax    -^  A^\d  -^  X) 

11.  Divide  -=— — r  bv  — , — 

a^  -\-  &    "   a  •\-  c 

12.  Divide  ■— by  ^-^^ — -—^^ 

X  "^      a  —  x 

,3.  Divide  ^qr-,^jq.-^  by  -^-^. 
14.  Divide  ^—  by  ^--^. 

iq.    Divide by  — ^^ -"     (See  Appendix,  p.  285.) 

■^  x-^-  ax      -^     I  ■\-  a 


CHAPTER    IX. 
SIMPLE     EQUATIONS. 

193.  An  Equation  is  an  expression  of  equality  between 
two  quantities.     (Art.  27.) 

194.  Every  equation  consists  of  two  parts,  called  the 
first  and  second  members. 

195.  The  First  Member  is  the  part  on  the  left  of  the 
sign  =. 

The  Second  Manber  is  the  part  on  the  right  of  the 
sign  =. 

196.  Equations  are  divided  into  degrees,  according  to  the 
exponent  of  the  unknown  quantity;  as  the  first,  second, 
third,  fourth,  etc. 

Equations  are  also  divided  into  Simple,  Quadratic, 
CuMc,  etc. 

197.  A  Simple  Equation  is  one  which  contains 
only  ihQ  first  power  of  the  unknown  quantity,  and  is  of  the 
first  degree ;  as,  ax  =  d. 

198.  A  Quadratic  Equation  is  one  in  which  the 
highest  power  of  the  unknown  quantity  is  a  square,  and  is 
of  the  second  degree  ;  as,  ax^  -\-  ex  ^  d. 

199.  A  Cubic  Equation  is  one  in  which  the  highest 
power  of  the  unknown  quantity  is  a  cuie,  and  is  of  the 
third  degree ;  as,  aa^  +  dx^  —  ex  =  d. 

193.  What  is  an  equation?  194.  How  many  parts?  195.  Which  is  the  flri«t 
member  ?  The  second  ?  196.  How  are  equations  divided  ?  What  other  divisions  ? 
197.  What  is  a  simple  equation  ?    198.  A  quadratic  ?    199.  Cubic  ? 


96  SIMPLE     EQUATIONS. 

200.  An  Identical  Equation  is  one  in  which  both 
members  have  the  same  form^  or  may  be  reduced  to  the 
same  form. 

Thus,   ab—c  =  ab—c,   and  8a;— 3a?  =  5a;,  are  identicaL 
Note. — Sucli  an  equation  is  often  called  an  identity/. 

201.  The  Transformation  of  an  equation  is  chang- 
ing its  form  without  destroying  the  equality  of  its  members. 

Note. — The  members  of  an  equation  will  retain  their  equality,  so 
long  as  they  are  equally  increased  or  diminished.     (Ax.  2-5.) 

TRANSPOSITION. 

202.  Transposition  of  Terms  is  changing  them 
from  one  side  of  an  equation  to  the  other  without  destroy- 
ing the  equahty  of  its  members. 

203.  Unknown  Quantities  may  be  combined  with 
known  quantities  by  addition,  subtraction,  multiplication, 
or  division. 

Note. — The  object  of  transposition  is  to  obtain  an  equation  in  which 
the  terms  containing  the  unknown  quantity  shall  stand  on  one  side, 
and  the  known  terms  on  the  other. 

204.  To    Transpose    a    Term   from    one    Member   of  an 

Equation  to  the  other. 

1.  Given  x  -{-1)  z=a,  to  find  the  value  of  x. 

Solution.— By  the  problem,  x+b  =  a 

Adding  —b  to  each  side  (Ax.  2),    x+b—b  =  a—b 
Cancelling  (  +  &—&)  (Ax.  7),  .*.     x  =  a—b 

This  result  is  the  same  as  changing  the  sign  of  b  from  +  to  —  in 
ihe  first  equation,  and  transposing  it  to  the  other  side. 

2.  Given  x  —  d  z=  c,  to  find  the  value  of  x. 

Solution. — By  the  problem,  x—d  =  c 

Adding  +dto  each  side  (Ax.  2),  x—d+d  =  c  +  d 
Cancelling  (—d  +  d),  (Ax.  7.)  .-.    x  =  c+d 

200.  Identical  ?  201.  What  is  the  transformation  of  an  equation  ?  Note.  Equality. 
202.  What  is  transposition  of  terms?  203.  How  combine  unknown  quantities T 
Note.  Object  of  transposition  ? 


ONE     UNKNOWN     QUANTITY.  97 

This  result  is  also  tlie  same  as  changing  the  sign  of  d  from  —  io  4- , 
and  transposing  it  to  the  other  side.     Hence,  the 

^vJjE.— Transpose  the  term  from  one  member  of  the  equa- 
tion to  the  other,  and  change  its  sign. 

Note.— In  the  first  of  the  preceding  examples,  the  unknown 
quantity  is  combined  with  one  that  is  known  by  addition;  in  the 
second,  with  one  by  subtraction. 

3.  Given  b  — c-{-x  =  a  —  d,  to  find  x, 

4.  Given  x-\-al)  — c=a-\-l),  to  find  x. 

205.  The  Signs  of  all  the  terms  of  an  equation  may  be 
changed  without  destroying  the  equality.  For,  all  the 
terms  on  each  side  may  be  transposed  to  the  other,  by 
changing  their  signs. 

206.  If  all  the  terms  on  one  side  are  transposed  to  the 
other,  each  member  will  be  equal  to  o. 

Thus,  if  05+6  =  <?,  it  follows  that  x+c—d  =  o. 


REDUCTION    OF    EQUATIONS. 

207.  Tlie  Heduction  of  an  equation  consists  in  finding 
the  value  of  the  unknown  quantity  which  it  contains. 

208.  The  Value  of  an  unknown  quantity  is  the  number 
which,  substituted  for  it,  will  satisfy  the  equation.  Hence, 
it  is  sometimes  called  the  root  of  the  equation. 

209.  The  reduction  of  equations  depends  on  the  following 

PRINCIPLE. 

Both  members  of  an  equation  may  be  increased  or  dimin- 
ished by  the  same  quantity  ivithout  destroying  the  equality. 

204.  How  transpose  a  term  from  one  member  of  an  equation  to  the  other? 
205.  What  is  the  effect  of  changing  all  the  signs  ?  206.  Of  transposing  all  the  terms  f 
207.  In  what  does  the  reduction  of  an  equation  consist  ?  208.  What  is  the  value  of 
an  nnkno\ATi  quantity  ?  What  sometimes  called  ?  209.  Upon  what  principle  does 
the  reduction  of  equations  depend  ? 


98 


SIMPLE     EQUATIONS 


210.  This  principle  may  be  illustrated  by  a  pair  ol 
scales.  If  4  balls,  each  weighing  i  lb.,  are  placed  in  each 
scale,  they  balance  each  other. 

Adding  2  lbs.  to  each  scale, 

4  +  2  =  4  +  2 

Subtracting  2  lbs.  from  each, 

4-2  =  4-2 
Multiplying  each  by  2, 

4x2  =  4x2 
Dividing  each  by  2, 

4-5-2  =  4-7-2 

211.  To    Reduce   an    Equation    containing   One   Unlcnown 

Quantity  by  Transposition, 

5.  Given  2a;  —  3^?  -f  7  =  a;  -f  35,  to  find  x. 

Solution.  —By  the  problem,  2a;— 3a  +  7  =  a; + 35 

Transposing  the  terms  (Art.  204),      2X—x  =  35— 7  +  3a 
Uniting  the  terms,     (Ax.  9),  Ans.x  =  2S  +  3a 

Therefore,  28  +  3a  is  the  value  of  x  required.    Hence,  the 

EuLE. — Transpose  the  unhnoivn  quantities  to  one  sid6, 
and  the  known  quantities  to  the  other,  and  unite  the  terms. 

Notes. — i.  Transposing  the  terms  is  the  same,  in  effect,  as  adding 
equal  quantities  to,  or  subtracting  them  from  each  member;  hence, 
it  is  often  called  reduction  of  equations  by  addition  or  svbtractiovu 

2.  If  the  unknown  quantity  has  the  sign  —  before  it,  change  the 
Bigns  of  all  the  terms.    (Art.  205.) 

212.  When  the  same  term,  having  the  same  sign,  is  on 
opposite  sides  of  the  equation,  it  may  be  cancelled.    (Ax.  3.) 

6.  Reduce  7,x  ■\-  a  ^  6  =  b  —  4  -^  2X. 

7.  Reduce  x— ■^■^c=z2X'\-a  —  d. 

8.  Reduce  21/  -{-  he  —  ad  =  y  +  2m  —  8. 

9.  Reduce  sab  —  y  -}-  d  =  —  2y-f  17. 

10.  Reduce  4cd  -f  27  —  4ic  -|-  J  =  28  —  3a:  -f  3M. 

11.  Reduce  5  -f  c  —  4a;  =  32  -f  &  —  $x  +  d. 

12.  Reduce  a:  -f  4  —  2a;  —  3  =  3a;  -f  4  -f  8  —  ^x. 

aio.  Illustrate  this  principle?  211.  What  is  the  rule  for  reducing  equations- ? 
Vote.  To  what  is  transposition  equivalent  ?  212.  When  the  same  term,  having  the 
fam^  sign,  is  on  opposite  sides,  what  may  be  done  ? 


ONE    unk:n^own    quantity.  99 


CLEARING     OF     FRACTIONS. 
213.  To  Reduce  an  Equation  containing  Fractions. 

1.  Given  -  +  -—  =  27,  to  find  the  value  of  x. 

Solution. — By  the  problem,  ?  +  ??==    2? 

2       6 
Multiplying  each  term  by  6,  the  I,  c.  ni, ) 

of  the  denominators  (Art.  148),  f  3^+ 22?  =  162 

Uniting  the  terms,  52?  =  162 

Dividing  each  side  by  the  coefficient,  Ans.  x  =    32I 

2.  Given  -  -\-  -  =  —,  to  find  the  value  of  x, 

23        4 

Solution.— By  the  problem,  ?  4.  ?  =  ^ 

234 
Mult,  by  12,  the  I,  c,  in.  of  denominators,  6a; +42;  =  90 

Uniting  terms,  and  dividing  (Art.  211),       Ans.  x=    g 

Therefore,  the  value  of  x  is  9.    Hence,  the 

KuLE. — Multiply  each  term  of  the  equation  ly  the  least 
common  multiple  of  the  denominators ;  then,  transposing 
and  imiting  the  terms,  divide  each  member  hy  the  coefficient 
of  the  unknown  quantity. 

Notes. — i.  An  equation  may  also  be  cleared  of  fractions,  by  multi- 
plying both  sides  by  each  denominator  separately. 

2.  The  reason  that  clearing  an  equation  of  fractions  does  not  destroy 
the  equation,  is  because  both  members  are  fnultipUed  by  the  same 
quantity.    (Ax,  4.) 

3.  A  fraction  is  multiplied  by  its  denominator  by  cancelling  the 
denominator.     (Art.  184,  Wote  i.) 

4.  Removing  the  coefficient  of  a  quantity  divides  the  quantity  by  it. 

5.  If  any  given  numerator  is  a  multiple  of  its  denominator,  divide 
the  former  by  the  latter  before  applying  the  rule. 

6.  The  unknown  quantity  in  the  last  two  problems  is  combined 
with  those  that  are  known  by  multiplication  and  division.  Hence,  the 
operation  is  often  called,  reduction  of  equations  by  multiplication  and 
division. 

213.  Rule  for  clearing  an  equation  of  fractions  ?  Notes,  i.  In  what  other  way  may 
fractions  be  removed  ?  2.  Why  does  not  this  process  destroy  the  equation  ?  3.  What 
:8  the  effect  of  cancelling  a  denominator?  4.  Effect  of  removing  a  coefficient' 
5.  If  a  numerator  is  a  multiple  of  its  denominator,  how  proceed  ? 


100  SIMPLE     EQUATIONS. 

3.  Eeduce  —  +  12  =  —  +  i. 

5  3  ; 

4.  Eeduce -  =  6x  —  66» 

3        6 

5.  Reduce  —  4--=  'zk  —x. 

10      5 

214.  When  the  sign  —  is  prefixed  to  a  fraction  and  the 
denominator  is  removed,  the  sigjis  of  all  the  terms  in  the 
numerator  must  be  changed  from  -f  to  — ,  or  —  to  +. 

X  —  2 

6.  Eeduce  30; =  20, 

Solution.— By  tlie  problem,  3a;  —  ^^  =   20 

Removing  tlie  denominator  5,  15a;— a; +  2  =  100 

Uniting  and  transposing  the  terms,  14a;  =    98 

Dividing  by  the  coeflacient,  Ans.  x—     7 

7.  Given  4^^::^^  =_  3^,  to  find  ^. 

X  d' 

8.  Given  ^x  —  —  =  a —^ ,  to  find  x, 

5  10 

9.  Given  — a;H [-^  =  -^,  to  find  x. 

3        4        24' 

10.  Given  «-|-5  +  c  =  --i [-^  +  ^+<?-^5^  to  find  rzr. 

2       4 

215.  The  principles  developed  by  the  preceding  illustra- 
tions may  be  summed  up  in  the  following 


GENERAL    RULE. 

I.  Clear  the  equation  of  fractions.     (Art.  213.) 

II.  Transpose  and  unite  tlie  terms.     (Art.  204.) 

III.  Divide  loth  sides  hy  the  coefficient  of  the  unknown 
quantity,     (Art.  213,  Note  4.) 

Proof. — For  the  unknoivn  quantity  substitute  its  value, 
and  if  it  satisfies  the  equation,  the  worlc  is  riglit. 

214.  If  sign  —  is  prefixed  to  a  fraction  ?    315,  What  is  the  general  rule  for  simple 
equations  ?    How  proved  ? 


OlJE     UNKNOWN     QUANTITY,  101 


EXAMPLES. 

1.  Given  x  ■] 1-  -  =  14,  to  find  x, 

2      4 

2.  Given  -  4-  a;  =  —  +  40,  to  find  x, 

2  10 

3.  Given  —  -f  10  =  —  +  13,  to  find  x. 

^  5  10        ^^ 

4.  Given  — - — \-  6  =  S,  to  find  x, 

^  X  —  2 

2X  -t-  I 

5.  Given  x  -\ =10,  to  find  x. 

6.  Given  2x  +  ^^±1L  =  18  +  9£r:29    ^^  ^^^  ^ 

5  4 

7.  Given  -  H ^  -  =  78,  to  find  x. 

234 

8.  Given  ^^^-8=^-^  +  5,  to  find  x. 

0  4 

9.  Given  x  +  ^^~^  +  '^^~     =  12,  to  find  a;. 

26 

10.  Given  2a;  —  16  =  ^^ ,  to  find  x. 

3 

^.         2a;  — 8    ,  a;  +  32   ,   ic  .     £   ;i 

11.  Given 5—^-  +  -  =  30,  to  find  x, 

12.  Given  -  +  -^=16  +  -,  to  find  a:. 

20  o 

rt-         3^+1  2a;   ,   a;—  I     ,     ^    , 

13.  Given —  10  = 1 — ,  to  find  x. 

2  30 

14.  Given f-  v  =  —  —  i  A^  to  find  x. 

^  10        6        15         ^^ 

15.  Given  Sx  +  6i  —  -  =  Si-  —  +  •^,  to  find  x. 

-^  ^2        ^72 

Note. — Sometimes  there  is  an  advantage  in  uniting  similar  terms, 
before  clearing  of  fractions.    Tims,  uniting  6^  witli  8^  ;  also witli 

— ,  we  nave,  8a;  =  2 +  8a; ;    /.  a;  =  7,  Ans. 


102  SIMPLE     EQUATIONS. 

i6.  Given  f -6 +^  =  1+ 2,  to  find  a;. 

^ .  AX         ZX 

17.  Given  —  =  —  +  15  —  12,  to  find  x. 

5        4 

18.  Given  20;  —  4  =  -  +  2,  to  find  x, 

2 

19.  Given   ^+  ^_  i|::^3^  ^  ,^j^   j^  g^^  ^ 

45  5 

X 

20.  Given  -  =  5  -}-  c,  to  find  x, 

(t 

fix 

21.  Given  —  =  ^Z,  to  find  x. 

^.        ax      hx  ,     n    -, 

22.  Given =  c,  to  find  x. 

23 

^.         2ax  -{-  h      ex  +  d    ^     „    ^ 

2$.  Given -^—  =  — 3_    to  find  x. 

a  c 

24.  Given h-  =  -  +  -,  to  find  x. 

a       X      2      a 

25.  Given  |+|  +  f  =  f  +  ^f-^  to  find  x. 

26.  Gi,en  ^  +'-^  =  '1  -^'J-^,  to  nnd  X, 

2         3        5         3 

^.         3flf  +  a;  6     ,     ^    - 

27.  Given  - — ■ 5  =-,  to  find  x, 

X  ^      x' 

X  —  I  I 

28.  Given  — ; h  i  =  -,  to  find  x. 

ic  +  I  a 

r^.  X    ,         X  a  ,       n     -, 

29.  Given  -  H =  -  +  «.  to  find  x. 

a      c  —■  a      c        ' 

a? 

30.  Given  x-{-h=z ,  to  find  x. 

31.  Given  x  —  a  =  — -^— - ,  to  find  x. 

X  —  a 

32.  Given  3  (^*)  +  p^-*)  =  4  (^*),  to  find  ^. 
3:>  Given  ^-~  =  x --,  to  find  x. 


OKE     UKKNOWK     QUANTITY.  103 

34.  Given  ~-\-  x =  25,  to  find  x. 

4  2 

X  X 

35.  Given  80  =  4a; ,  to  find  x. 

2       6 

,^.  2iC+I  a;^^ 

36.  Given  ! —  =  2X -^-^ ,  to  find  x, 

3  4 

37.  Given  10  ->  22;  =  5^+^  -  £1Z136    ^^  g^^  ^^ 

3  3 

38.  Given  a;  —  3  =  15  _  ^Jli,  to  find  x. 

39.  Given  a;  +  2  =  3a;  +  ^-±-?  -  ?-±-^,  to  find  a?. 

4  3 

^.         Z^  ,   X  —  4      a;  —  10 

40.  Given  ^  + 5 _  ^  _  5^^  g^^ 

422 

^.         iia;— I       ex— 11       x—i 

41.  Given  — - —  =  ^ to  find  x. 

12  4  10 

42.  Given  ^  _  ??  =  120,  to  find  x. 

5        10 

43.  Given  ar  —  20  =  —  ^^+1    to  find  x, 

5 

44.  Given  ~^  =  ^—^  ■\-t2-x,  to  find  x, 

^  3 

45.  Given  ^-^  +  10  =  —  _  1^=^,  to  find  x. 

46.  Given  — ^  -  -^-  =  5,  to  find  a:. 

I*  -j-    I  Q,  I 

r\'  X  2  +  X  C 

47.  Given  ^-^  -  ^_  =.____,  to  find  x. 

48.  Given  ^^=5+^,  to  find  ^. 

49-  Given  —  =  ^  — — ,  to  find  a?. 
2  3 

50.  Given  8«=  ^1^,  to  find  x, 

1  -\-  X 

51.  Given  -~^-^-  =  ~^yto  find  a;. 


104:  SIMPLE     EQUATIONS. 


PROBLEMS. 

216.  The  Solution  of  a  problem  is  finding  a  quantity 
which  will  satisfy  its  conditions.    It  consists  of  two  parts  : 

First.— The  Formation  of  an  equation  which  will 
express  the  conditions  of  the  problem  in  algebraic  language, 
Second. — The  deduction  of  this  equation. 

217.  To  Solve  Problems  in   Simple   Equations  containing 

one  unknown  Quantity. 

I.  A  farmer  divided  52  apples  among  3  boys  in  such  a 
manner  that  B  had  i  half  as  many  as  A,  and  C  3  fourths  as 
many  as  A  minus  2.     How  many  had  each  ? 

I.  Formation— Let  x  =  A'a  number. 

By  the  conditions,  -  =  B's        ** 

2 

4 

Therefore,  by  Ax.  0,        x  +  -  +  —  —  2=    52,  the  whole. 
24 
2c  Reduction—  4a;+2a;+3«— 8  =  208 

Transposing,  etc.,  gx  =  216 

Removing  the  coefficient,  x  =    24,  A's  number. 

Ans.  A  had  24,  B  had  12,  and  C  had  18—2  =  16. 

From  this  illustration  we  derive  the  following 

GENERAL     RULE. 

I.  Represent  the  unknown  quantity  ly  a  letter,  then  state 
in  algelraic  language  the  operations  necessary  to  satisfy  the 
wnditions  of  the  prohlem. 

II.  Clear  the  equation  of  fractions  ;  then,  transposing  and 
uniting  the  terms,  divide  each  member  by  the  coefficient  of  the 
unhnown  quantity.     (Art.  213.) 

Note. — A  careful  study  of  the  conditions  of  the  problem  will  soon 
enable  the  learner  to  discover  the  quantity  to  be  represented  by  the 
letter,  and  the  method  of  forming  the  equation. 

tx6.  What  is  the  Bolution  of  a  problem  ?  Of  what  does  it  coneist  ?  217.  What  is 
the  general  rale  ? 


OKE     TJKKKOWiq-     QTJAI^TITY.  105 

2.  The  bill  for  a  coat  and  vest  is  I40 ;  the  value  of  the 
coat  is  4  times  that  of  the  vest.     What  is  the  value  of  each  ? 

3.  A  bankrupt  had  $9000  to  pay  A,  B,  and  C ;  he  paid  B 
twice  as  much  as  A,  and  C  as  much  as  A  and  B.  What 
did  each  receive  ? 

4.  The  whole  number  of  hands  employed  in  a  factory  was 
1000  ;  there  were  twice  as  many  boys  as  men,  and  11  times 
as  many  women  as  boys.     How  many  of  each  ? 

5.  Two  trains  start  at  the  same  time,  at  opposite  ends  of 
a  railroad  1 20  miles  long,  one  running  twice  as  fast  as  the 
other.     How  far  will  each  have  run  at  the  time  of  meeting  ? 

6.  A  man  bought  equal  quantities  of  two  kinds  of  flour, 
at  $10  and  $8  a  barrel.  How  many  barrels  did  he  buy,  the 
whole  cost  being  1 1200  ? 

7.  If  96  pears  are  divided  among  3  boys,  so  that  the 
second  shall  have  2,  and  the  third  5,  as  often  as  the  first 
has  I,  how  many  will  each  receive  ? 

8.  A  post  is  one-fourth  of  its  length  in  the  mud,  one- 
third  in  the  water,  and  12  feet  above  water;  what  is  its 
whole  length  ? 

9.  After  paying  away  I  of  my  money,  and  then  J  of  the 
remainder,  I  have  $72.     What  sum  had  I  at  first  ? 

10.  Divide  $300  between  A,  B,  and  0,  so  that  A  may 
have  twice  as  much  as  B,  and  C  as  much  as  both  the  others. 

1 1 .  At  the  time  of  marriage,  a  man  was  twice  as  old  as 
his  wife;  but  after  they  had  lived  together  18  years,  his  age 
was  to  hers  as  3  to  2.     Eequired  their  ages  on  the  wedding  day. 

12.  A  and  B  invest  equal  amounts  in  trade.  A  gains 
1 1 260  and  B  loses  $870;  A's  money  is  now  double  B's. 
What  sum  did  each  invest  ? 

13.  Eequired  two  numbers  whose  difference  is  25,  and 
twice  their  sum  is  114. 

14.  A  merchant  buying  goods  in  New  York,  spends  the 
first  day  ^  of  his  money ;  the  second  day,  I ;  the  third  day, 
I;  the  fourth  day,  ^;  and  he  then  has  $300  left.  How 
much  had  he  at  first  ? 


106  SIMPLE    EQUATIOKS. 

15.  What  number  is  that,  from  the  triple  of  which  if  17 
be  subtracted  the  remainder  is  22  ? 

16.  In  fencing  the  side  of  a  field  whose  length  was  450 
rods,  two  workmen  were  employed,  one  of  whom  built  9  rods 
and  the  other  6  rods  per  day.  How  many  days  did  they 
work  ? 

17.  Two  persons,  420  miles  apart,  take  the  cars  at  the 
same  time  to  meet  each  other ;  one  travels  at  the  rate  of  40 
miles  an  hour,  and  the  other  at  the  rate  of  30  miles.  What 
distance  does  each  go  ? 

18.  Divide  a  line  of  28  inches  in  length  into  two  such 
parts  that  one  may  be  J  of  the  other. 

19.  Charles  and  Henry  have  $200,  and  Charles  has  seven 
times  as  much  money  as  Henry.     How  much  has  each  ? 

20.  What  is  the  time  of  day,  provided  f  of  the  time  past 
midnight  equals  the  time  to  noon  ? 

21.  A  can  plow  a  field  in  20  days,  B  in  30  days,  and  C  in 
40  days.     In  what  time  can  they  together  plow  it  ? 

22.  A  man  sold  the  same  number  of  horses,  cows,  and 
sheep;  the  horses  at  lioo,  the  cows  at  $45,  and  the  sheep 
at  I5,  receiving  $4800.     How  many  of  each  did  he  sell  ? 

23.  Divide  150  oranges  among  3  boys,  so  that  as  often  as 
the  first  has  2,  the  second  shall  have  5,  and  the  third  3. 
How  many  should  each  receive  ? 

24.  Four  geese,  three  turkeys,  and  ten  chickens  cost  lio  ; 
a  turkey  cost  twice  as  much  as  a  goose,  and  a  goose  3  times 
as  much  as  a  chicken.    What  was  the  price  of  each  ? 

25.  The  head  of  a  fish  is  4  inches  long ;  its  tail  is  1 2  times 
as  long  as  its  head,  and  the  body  is  one-half  the  whole 
length.     How  long  is  the  fish  ? 

26.  Divide  100  into  two  parts,  such  that  one  shall  be  20 
more  than  the  other. 

27.  Divide  a  into  two  such  parts,  that  the  greater  divided 
by  c  shall  be  equal  to  the  less  divided  by  d. 

28.  How  much  money  has  A,  if  \,  f ,  and  f  of  it  amount 
to  $1222  ? 


OKE     UHKHOWN     QUANTITY.  107 

29.  What  number  is  that,  |,  J,  J,  and  J  of  which  are 
equal  to  60  ?  * 

30.  A  man  bought  beef  at  25  cents  a  pound,  and  twice  as 
much  mutton  at  20  cents,  to  the  amount  of  $39.  How 
many  pounds  of  each  ? 

31.  A  says  to  B,  "I  am  twice  as  old  as  you,  and  if  I  were 
15  years  older,  I  should  be  3  times  as  old  as  you."  What 
were  their  ages  ? 

32.  The  sum  of  the  ages  of  A,  B,  and  C  is  1 10  years ;  B  is 
3  years  younger  than  A,  and  5  years  older  than  C.  What 
are  their  ages  ? 

^^.  At  an  election,  the  successful  candidate  had  a 
majority  of  150  votes  out  of  2500.  What  was  his  number 
of  votes  ? 

34.  In  a  regiment  containing  1200  men,  there  were 
3  times  as  many  cavalry  as  artillery  less  20,  and  92  more 
infantry  than  cavalry.    How  many  of  each  ? 

35.  Divide  $2000  among  A,  B,  and  0,  giving  A  $100 
more  than  B,  and  $200  less  than  0.  What  is  the  share  of 
each  ? 

36.  A  prize  of  $150  is  to  be  divided  between  two  pupils, 
and  one  is  to  have  |  as  much  as  the  other.  What  are  the 
shares  ? 

*  When  the  conditions  of  the  problem  contain  fractional  expressions, 
as  ^,  J,  \,  etc. ,  we  can  avoid  these  fractions,  and  greatly  abridge  the 
operation,  by  representing  the  quantity  sought  by  such  a  number  of 
ic's  as  can  be  divided  by  each  of  the  denominators  without  a  remainder. 
This  number  is  easily  found  by  taking  the  least  common  multiple  of 
all  the  denominators.     Thus,  in  problem  29, 

Let  \2X  =  the  number. 

Then  will  6x  —  i  half. 

«        "  4a;  =  I  third. 

**        "  3iC  =  I  fourth. 

"        '*  225—1  sixth. 

Hence,  6a; + 4a! + 3a;  +  2a?  =  60 

.'.   a;  =  4 
Finally,  a;x  12  or  12a!  =  48,  the  number  required 


108  SIMPLE     EQUATIONS. 

37.  Two  horses  cost  16 16,  and  5  times  the  cost  of  one  was 
6  times  the  cost  of  the  other.     What  was  the  price  of  each  ? 

38.  What  were  the  ages  of  three  brothers,  whose  united 
ages  were  48  years,  and  their  birthdays  2  years  apart  ? 

39.  A  messenger  travelling  50  miles  a  day  had  been  gone 

5  days,  when  another  was  sent  to  overtake  him,  travelling 
65  miles  a  day.     How  many  days  were  required  ? 

40.  What  number  is  that  to  which  if  75  be  added,  f  of 
the  sum  will  be  250  ? 

41.  It  is  required  to  divide  48  into  two  parts,  which  shall 
be  to  each  other  as  5  to  3.* 

42.  What  quantity  is  that,  the  half,  third,  and  fourth  of 
which  is  equal  to  a  ? 

43.  A  and  B  together  bought  540  acres  of  land,  and 
divided  it  so  that  A's  share  was  to  B's  as  5  to  7.  How  many 
acres  had  each  ? 

44.  A  cistern  has  3  faucets;   the  first  will  empty  it  in 

6  hours,  the  second  in  10,  and  the  third  in  12  hours.     How 
long  will  it  take  to  empty  it,  if  all  run  together  ? 

45.  Divide  the  number  39  into  4  parts,  such  that  if  the 
first  be  increased  by  i,  the  second  diminished  by  2,  the  third 
multiplied  by  3,  and  the  fourth  divided  by  4,  the  results 
will  be  equal  to  each  other. 

46.  Find  a  number  which,  if  multiplied  by  6,  and  1 2  be 
added  to  the  product,  the  sum  will  be  66. 

47.  A  man  bought  sheep  for  $94  ;  having  lost  7  of  them, 
he  sold  ^  of  the  remainder  at  cost,  receiving  $20.  How 
many  did  he  buy  ? 

48.  A  and  B  have  the  same  income ;  A  saves  J  of  his,  but 
B  spending  $50  a  year  more  than  A,  at  the  end  of  5  years  is 
$100  in  debt.    What  is  their  income  ? 

*  When  the  quantities  sought  have  a  given  ratio  to  each  other,  the 
solution  may  be  abridged  by  taking  such  a  number  of  a's  for  the 
unknown  quantity,  as  will  express  the  ratio  of  the  quantities  to  each 
other  without  fractions.  Thus,  taking  5a;  for  the  first  part,  2X  will 
represent  the  second  part ;  then  5 j;  +  3a;  =  48,  etc. 


ONE     UKKITOWK     QUAKTITY.  109 

49.  A  cistern  is  supplied  with  water  by  one  pipe  and 
emptied  by  another;  the  former  fills  it  in  20  minutes,  the 
latter  empties  it  in  15  minutes.  When  full,  and  both  pipes 
run  at  the  same  time,  how  long  will  it  take  to  empty  it  ? 

50.  What  number  is  that,  if  multiplied  by  vi  and  n 
separately,  the  difference  of  their  products  shall  be  c?  ? 

51.  A  hare  is  50  leaps  before  a  greyhound,  and  takes 
4  leaps  to  the  hound's  3  leaps ;  but  2  of  the  greyhound's 
equal  3  of  the  hare's  leaps.  How  many  leaps  must  the 
hound  take  to  catch  the  hare  ? 

52.  What  two  numbers,  whose  difference  is  h,  are  to  each 
other  as  «  to  c  ? 

53.  A  fish  was  caught  whose  tail  weighed  9  lbs.;  his  head 
weighed  as  much  as  his  tail  and  half  his  body,  and  his  body 
weighed  as  much  as  his  head  and  tail  together.  What  was 
the  weight  of  the  fish  ? 

54.  An  express  messenger  travels  at  the  rate  of  13  miles 
in  2  hours;  12  hours  later,  another  starts  to  overtake  him, 
travelling  at  the  rate  of  26  miles  in  3  hours.  How  long  and 
how  far  must  the  second  travel  before  he  overtakes  the  first  ? 

55.  A  father's  age  is  twice  that  of  his  son  ;  but  10  years 
ago  it  was  3  times  as  great.     What  is  the  age  of  each  ? 

56.  What  number  is  that  of  which  the  fourth  exceeds  the 
seventh  part  by  30  ? 

57.  Divide  $576  among  3  persons,  so  that  •  the  first  may 
have  three  times  as  much  as  the  second,  and  the  third  one- 
third  as  much  as  the  first  and  second  together. 

58.  In  the  composition  of  a  quantity  of  gunpowder,  the 
nitre  was  10  lbs.  more  than  f  of  the  whole,  the  sulphur 
4 J  lbs.  less  than  J  of  the  whole,  the  charcoal  2  lbs.  less  than 
\  of  the  nitre.     What  was  the  amount  of  gunpowder  ?  * 

*  The  operation  will  be  ^ortened  by  the  following  artifice  : 

Let  42a;  +  48  =  the  number  of  pounds  of  powder. 

Then  28a; +  42  =  nitre  ;  jx+s^  =  sulphur  ;  4X  +  4  =  charcoal. 

Hence,   390;  + 49 ^  =  42a;  +  48. 

.'.  X 


110  •  SIMPLE     EQUATIONS. 

59.  Divide  $6  into  3  parts,  sucli  that  J  of  the  first,  J  of 
the  second,  and  ^  of  the  third  are  all  equal  to  each  other. 

60.  Divide  a  line  21  inches  long  into  two  parts,  such  that 
one  may  be  |  of  the  other. 

61.  A  milliner  j)aid  $5  a  month  foi  rent,  and  at  the  end 
of  each  month  added  to  that  part  of  1  er  money  which  was 
not  thus  spent  a  sum  equal  to  i  half  of  this  part;  at  the 
end  of  the  second  month  her  original  money  was  doubled. 
How  much  had  she  at  first  ? 

62.  A  man  was  hired  for  60  days,  on  condition  that  for 
every  day  he  worked  he  should  receive  75  cents,  and  for 
every  day  he  was  absent  he  should  forfeit  25  cents;  at  the 
end  of  the  time  he  received  $12.  How  many  days  did  he 
work? 

6^.  Divide  $4200  between  two  persons,  so  that  for  every 
I3  one  received,  the  other  shall  receive  $5. 

64.  A  father  told  his  son  that  for  every  day  he  was  perfect 
in  school  he  would  give  him  15  cents;  but  for  every  day  he 
failed  he  should  charge  him  10  cents.  At  the  end  of  the 
term  of  1 2  weeks,  60  school  days,  the  boy  received  $6.  How 
many  days  did  he  fail  ? 

65.  A  young  man  spends  ^  of  his  annual  income  for 
board,  -|  for  clothing,  -^  in  charity,  and  saves  I318.  What 
is  his  income  ? 

66.  A  certain  sum  is  divided  so  that  A  has  $30  less  than  ^, 
B  lio  less  than  |,  and  0  $8  more  than  J  of  it.  What  does 
each  receive,  and  what  is  the  sum  divided  ? 

67.  The  ages  of  two  brothers  are  as  2  to  3 ;  four  years 
hence  they  will  be  as  5  to  7.    What  are  their  ages  ?  * 

Note. — To  change  a  proportion  into  an  equation,  it  is  necessary  to 
assume  the  truth  of  the  following  well  established  principle : 

If  four  quantities  are  proportional,  the  product  of  the  extremes  is 

*  A  strict  conformity  to  system  would  require  that  this  and- similar 
problems  should  be  placed  after  the  subject  of  proportion ;  but  it  is 
convenient  for  the  learner  to  be  able  to  convert  a  proportion  into  an 
equation  at  this  stage  of  his  progress. 


ONE     UNKNOWN^     QUANTITY-.  Ill 

equal  to  the  product  of  fhe  means.  Hence,  in  such  cases,  we  have  only 
to  make  the  product  of  the  extremes  one  side  of  the  equation,  and  the 
product  of  the  means  the  other. 

Thus,  let  2X  and  3a;  be  equal  to  their  respective  ages. 

Then  2a;+4  :  3a'  +  4  -  5  :  7- 

Making  the  product  of  the  extremes  equal  to  the  product  of  the 

mean^  we  have, 

140;+ 28  =  150;+ 20. 

Transposing,  uniting  terms,  etc.,  a;  =  8. 

.'.  2X=  16,  the  younger ;  and  3a;  =  24,  the  older. 

68.  What  two  numbers  are  as  3  to  4,  to  each  of  which  if 
4  be  added,  the  sums  will  be  as  5  to  6  ? 

69.  The  sum  of  two  numbers  is  5760,  and  their  difference 
is  equal  to  J  of  the  greater.     What  are  the  numbers  ? 

70.  It  takes  a  college  crew  which  in  still  water  can  pulLat 
the  rate  of  9  miles  an  hour,  twice  as  long  to  come  up  the 
river  as  to  go  down.     At  what  rate  does  the  river  flow  ? 

71.  One-tenth  of  a  rod  is  colored  red,  ^  orange,  -^  yellow, 
-}^  green,  -f^  blue,  ^^  indigo,  and  the  remainder,  302  inches, 
violet.     What  is  its  length  ? 

72.  Of  a  certain  dynasty,  |  of  the  kings  were  of  the  same 
name,  J  of  another,  J  of  another,  ^^  of  another,  and  there 
were  5  kings  besides.     How  many  were  there  of  each  name  ? 

73.  The  difference  of  the  squares  of  two  consecutive 
numbers  is  15.    What  are  the  numbers? 

74.  A  deer  is  80  of  her  own  leaps  before  a  greyhound ; 
she  takes  3  leaps  for  every  2  that  he  takes,  but  he  covers  as 
much  ground  in  one  leap  as  she  does  in  two.  How  many 
leaps  will  the  deer  have  taken  before  she  is  caught  ? 

75.  Two  steamers  sailing  from  New  York  to  Liverpool,  a 
distance  of  3000  miles,  start  from  the  former  at  the  same 
time,  one  making  a  round  trip  in  20  days,  the  other  in 
25  days.  How  long  before  they  will  meet  in  New  York, 
and  how  far  will  each  have  sailed  ? 

("See  Appendix,  p.  286.) 


CHAPTEE    X. 
SIMULTANEOUS     EQUATIONS. 

TWO    UNKNOWN    QUANTITIES. 

218.  Simultaneous  *  Equations  consist  of  two  oi 
more  equations,  each  containing  two  or  more  unk7iown 
quantities.  They  are  so  called  because  they  are  satisfied  by 
the  same  values. 

Thus,  x  +  y^'j  and  5a?— 4y  =  8  are  simultaneous  equations,  for  in 
each  aj  =  4  and  y  =  3. 

219.  Indepe^ident  Equations  are  those  which 
express  different  conditions,  so  that  one  cannot  be  reduced 
to  the  same  form  as  the  other. 

Thus,  6aj— 4^=14  and  23^  +  3^=:  22  are  independent  equations. 
But  the  equations  ic  +  y  =  5  and  3a;  +  3^  =  15  are  not  independent,  for 
one  is  directly  obtained  from  the  other.  Such  equations  are  termed 
dependent. 

Note. — Simultaneous  equations  are  usuaWj  independent ;  but  inde- 
pendent equations  may  not  be  simultaneous  ;  for  the  letters  employed 
may  have  the  same  or  different  values  in  the  respective  equations. 

Thus,  the  equations  a?+y=7  and  2a?— 2^=14  are  Independent, 
but  not  simultaneous ;  for  in  one  x  =  7—y,  in  the  other  x=  7+y,  etc. 

220.  Problems  containing  more  than  one  unknown  quan- 
tity must  have  as  mani/  simultaneous  equations  as  there  are 
unknown  quantities. 

If  there  are  more  equations  than  unknown  quantities, 
some  of  them  will  be  superfluous  or  contradictory. 

218.  What  are  Bimultaneous  equations  ?  219.  Independent  equations  ?  220.  Ho^ 
many  equations  must  each  problem  have  ? 

*  From  the  Latin  ^i'n^id,  at  the  same  time. 


TWO     UNKN^OWN     QUAl^TITIES.  113 

If  the  number  of  equations  be  less  than  the  number  of 
unknown  quantities,  the  problem  will  not  admit  of  a  definite 
answer,  and  is  said  to  be  indeterminate. 

221.  JSllmination*  is  combining  two  equations 
which  contain  two  unknown  quantities  into  a  single  equa- 
tion, having  but  one  unknown  quantity.  There  are  three 
methods  of  elimination,  viz. :  by  Comparison,  by  Substitution. 
and  by  Addition  or  Subtraction, 


CASE    I. 
222.  To  Eliminate  an  Unknown  Quantity  by  Comparison, 

I.  Given  x  -\-  y  =z  i6,  and  a;  — «/  =  4,  to  find  x  and  y. 

Solution.— By  the  problem, 
«  ** 

Transposing  the  ^  in  (i), 
"  the  7/  in  (2), 

By  Axiom  i. 
Transposing  and  uniting  terms. 

Substituting  the  value  of  y  in  (4), 


x+p=  16 

(I) 

x-y=   4 

(2) 

X  =  it-y 

(3) 

x=   A+y 

(4) 

4+y=  it—y 

(5) 

2y=  12 

(6) 

.-.   y=  6 

aj=  10 

In  (5)  it  will  be  seen  we  have  a  new  equation  which  contains 
only  one  unknown  quantity.  This  equation  is  reduced  in  the  usual 
way.     Hence,  the 

EuLE. — I.  From  each  equation  find  the  value  of  the  quan- 
tity to  he  eliminated  in  terms  of  the  other  quantities. 

II.  Form  a  new  equation  from  these  equal  values,  and 
reduce  it  ly  the  preceding  rules. 

Note. — This  rule  depends  upon  the  axiom,  that  things  which  are 
equal  to  the  same  thing  are  equal  to  each  other,    (Ax.  i.) 


For  convenience  of  reference,  the  equations  are  numbered  (i), 

(2),  (3),  (4),  etc. 

• 

221.  What  is  elimination?    Name  the  methods.    222.  How  eliminate  an  unknown 
ttuantity  by  comparison  ?    Note.  Upon  what  principle  does  this  rule  depend  ? 

*  From  the  Latin  eliminare,  to  cast  out. 


114  SIMPLE     EQUATIONS. 

2.  Given  x  -\-  y  =  12,  and  x—y-ir^  z=  8,  to  find  x  and  y 

3.  Given  3X-]-2y  =  48,  and  2x—$y  =  6,  to  find  x  and  y. 

4.  Given  x-\-y  =  20,  and  2:^+3?/  =  42,  to  find  x  and  y. 

5.  Given  4^+32/  =  i3>  and  3^^  +  21/  =  9,  to  find  x  and  ?/. 

6.  Given  30; +2?/ =118,  and  a; +5?/= 191,  to  find  a;  and  ^^'. 

7.  Given  4x-\-^y  =  22,  and  72:4-32^=27,  to  find  x  and  2/. 

CASE    II. 
223.  To  Eliminate  an  Unknown  Quantity  by  Substitution. 

8.  Given  x-{-2y  =  10,  and  ^x-^2y  z=  18,  to  find  x  and  y. 

Solution. — B7  the  problem,  x+2i/=io  (i) 

"  3a;+22/  =  i8  (2) 

Transposing  2y  in  (i),         .  x  =  10— 2y  (3) 

Substituting  tlie  value  of  x  in  (2),  30—4^  =  18  (4) 

Transposing  and  uniting  terms  (Art.  211),        42/  =  12  (5) 

'-   y=  3 
Substituting  the  value  of  2^  in  (i),  a;  =  4 

5^"  For  convenience,  we  first  find  the  value  of  the  letter  which  is 
least  involved.     Hence,  the 

Rule. — I.  From  one  of  the  equations  find  the  value  of  the 
unknown  quantity  to  he  eliminated,  in  terms  of  the  other 
quantities. 

II.  Substitute  this  value  for  the  same  quantity  in  the 
other  equation,  and  reduce  it  as  before. 

Notes. — i.  This  method  of  elimination  depends  on  Ax.  i. 
2.  The  given  equations  should  be  cleared  of  fractions  before  com- 
mencing the  elimination. 

9.  Given  x  +  ^y  =  19,  and  $x — 2y  =  10,  to  find  x  and  y. 

10.  Given  -  +  ^  =  7,  and  -  -|-  ^  =  8,  to  find  x  and  11. 

23'  3^2'  ^ 

11.  Given  2x-{-^y  =z  28,  and  30;+  2^  =  27,  to  find  x  and  y. 

12.  Given  4X-{-y  =  43,  and  5^+2^  =  56,  to  find  x  and  y. 

13.  Given  s^+^  =  7^>  and  5?/ +  32  =  'jx,  to  find  x  and  y. 

14.  Given  4X-]-sy=  22,  and  yx+^y  =  2^,  to  find  x  and  y. 

223.  How  eliminate  an  unknown  quantity  by  substitution?  If^ote  Upon  what 
principle  docs  this  method  depend? 


TWO     UNKNOWN     QUANTITIES.  115 


CASE     III. 

224.  To   Eliminate   an   Unknown  Quantity  by  Addition  or 
Subtraction, 

15.  Given  4a; +  3^  =18,  and  5a:— 2?/=:  11,  to  find  x  and  y. 

Solution. — By  tlie  problem,  4a; +  3^=18  (i) 

"  "  e,x—2y~  II  (2} 

Multiplying  (i)  by  2,  the  coef.  of  y  in  (2),    8a5+ 6y  =  36  (3) 

Multiplying  (2)  by  3,  the  coef.  of  y  in  (i),   isx—6y  =  33  (4) 

Adding  (3)  and  (4)  cancels  6y,  23a;         =  69  (5) 

,',    X         =3 
Substituting  the  value  of  a;  in  (i),  12  +  3^  =  18 

y=    2 

^°  In  the  preceding  solution,  y  is  eliminated  by  addition, 

16.  Given  6a: +  5?/ =2 8,  and  8a; +  3?/— 30,  to  find  x  and  y. 

Solution. — By  the  problem,  6a;  +  SV  =   28  (1) 

"            '*  8a;  +  3y=   30  (2) 

Multiplying  (i)  by  8,  the  coef.  of  a;  in  (2),  48a; +  402/  =  224  (3) 

Multiplying  (2)  by  6,  the  coef.  of  x  in  (i),  48a;  +  i8y  =  180  (4) 

Subtracting  (4)  from  (3),  22^  =    44 

.'.  y=    2 

Substituting  the  value  of  y  in  (2)  8a;+6  =   30 

.'.    x=     3 

B^  In  this  solution,  x  is  eliminated  by  subtraction.    Hence,  the 

EuLE. — I.  Select  the  letter  to  he  eliminated;  then  multiply 
or  divide  one  or  doth  equations  hy  such  a  number  as  ivill 
make  the  coefficients  of  this  letter  the  same  in  loth,   (Ax.  4,  5.) 

II.  If  the  signs  of  these  coefficients  are  alilce,  subtract  one 
equation  from  the  other  ;  if  unlihe,  add  the  two  equations 
together.    (Ax.  2,  3.) 

Notes. — i.  The  object  of  multiplying  or  dividing  the  equations  is  to 
equalize  the  coefficients  of  the  letter  to  be  eliminated. 

2.  If  the  coefficients  of  the  letter  to  be  eliminated  are  prime  num- 
bers, or  ^rime  to  each  other,  multiply  each  equation  by  the  coefficient 
of  this  letter  in  the  other  equation,  as  in  Ex.  15. 

224.  What  is  the  rule  for  elimination  by  addition  or  snbtraction  ?    Notes. — i.  The 
object  of  multiplying  or  dividing  the  equalion  ?    2.  If  coefficients  are  prime  ? 


116  SIMPLE     EQUATIONS. 

3.  If  not  prime,  divide  the  I.  c,  in*  of  the  coefficients  of  the  lett^i 
to  be  eliminated  by  each  of  these  coefficients,  and  the  respective 
quotients  will  be  the  multipliers  of  the  corresponding  equations. 
Thus,  the  I,  c,  in.  of  6  and  8,  the  coefficients  of  a;  in  Ex.  16,  is  24; 
hence,  the  multipliers  would  be  3  and  4. 

4.  If  the  coefficients  of  the  letter  to  be  eliminated  have  common 
factors,  the  operation  is  shortened  by  cancelling  these  factors  before 
the  multiplication  is  performed.  Thus,  by  cancelling  the  common 
factor  2  from  6  and  8,  the  coefficients  of  x  in  the  last  example,  they 
become  3  and  4,  and  the  labor  of  finding  the  I,  c,  in.  is  avoided. 

17.  Given  3a; +4?/ =2 9,  and  'jx-\-iiy=j6,to  find  a: and y. 

18.  Given  9a;— 4^=8,  and  13^-1- 72/==  loi,  to  find  ir  and  ^. 

19.  Given  ^x—'jy='jy  and  12a: +  5^/^94,  to  find  x  and  y. 

20.  Given  3^^+21/=  118,  and  0:4-5^=191,  to  find  xandy. 

21.  Given  4a; -{-52/= 22,  and  7a; +  32/= 27,  to  find  x  and  y. 

Rem. — The  preceding  methods  of  elimination  are  applicable  to  all 
simultaneous  simple  equations  containing  two  unknown  quantities, 
and  either  may  be  employed  at  the  pleasure  of  the  learner. 

The  first  method  has  the  merit  of  clearness,  but  often  gives  rise  to 
frax^tions. 

The  second  is  convenient  when  the  coefficient  of  one  of  the  unknown 
quantities  is  i ;  if  more  than  i,  it  is  liable  to  produce /7•ac^^07^«. 

The  third  never  gives  rise  to  fractions,  and,  in  general,  is  the  most 
simple  and  expeditious. 

EXAMPLES. 

Find  the  values  of  x  and  y  in  the  following  equations : 

1.  2a;  +    3^=    23,  5.       5a;  +    7^=   43, 
5a; —    2^=    10.  \ix -\-    9?/=    69. 

2.  4a;  +      y—    Z^,  6.       8a;  — 2iy=    n, 
4y+      x=    16.  6a: +  357/=  177. 

3.  30a;  +  4oy  =  270,  7.     21?^  +  20a;  =  165, 
50a;  4-  30^  =  340.  772^  —  30^  =  295. 

4.  2a;  +    jyzzz    34,  8.     I  la;  —  loy  =    14, 
SX+    gy=    51.  s^+    iy=    41. 

Notes. — 3.  If  not  prime,  how  proceed  ?  4.  If  the  coefficients  have  comuion  feo- 
tors,  how  shorten  the  operation  ? 


TWO     UNKNOWK     QUANTITIES.  ll? 


lO. 


II. 


12. 


13- 


14. 


6y~ 

2a;  =  208, 

15. 

Sx  + 

y  = 

'42, 

loa;  — 

42/  =  156. 

2X  + 

42/  = 

18. 

4X  + 

Sy=    22, 

16. 

2X  + 

42/  = 

20, 

5^- 

72/=     6. 

4X  + 

52/  = 

28. 

SX- 

5«/=    i3> 

17. 

4X-\- 

32/  = 

50, 

2X  + 

7^=    81. 

3«- 

32/  = 

6c 

5^- 

72/=    33j 

18. 

3^  + 

52/  = 

57, 

iia:  + 

121/  =  100. 

5«  + 

32/  = 

47. 

i* 

|=.8, 

19. 

;  + 

2(  _ 
3 

7> 

re 
2 

^    =    21. 

4 

i- 

4 

5. 

162;  + 

17?/ =  500, 

20. 

2a;  + 

2/    = 

50, 

17a;  — 

3«/=  no. 

i* 

^    _ 

7 

5. 

PROBLEMS. 

1.  Required  two  numbers  whose  sum  is  70,  and  whose 
difference  is  16. 

2.  A  boy  buys  8  lemons  and  4  oranges  for  56  cents;  and", 
afterwards  3  lemons  and  8  oranges  for  60  cents.  What  did 
he  pay  for  each  ? 

3.  At  a  certain  election,  375  persons  voted  for  two  candi- 
dates, and  the  candidate  chosen  had  a  majority  of  91.  How 
many  voted  for  each  ? 

4.  Divide  the  number  75  into  two  such  parts  that  three 
times  the  greater  may  exceed  seven  times  the  less  by  15. 

5.  A  farmer  sells  nine  horses  and  seven  cows  for  I1200; 
and  six  horses  and  thirteen  cows  for  an  equal  amount. 
What  was  the  price  of  each  ? 

6.  From  a  company  of  ladies  and  gentlemen,  15  ladies 
retire;  there  are  then  left  two  gentlemen  to  each  lady. 
After  which  45  gentlemen  depart,  when  there  are  left  five 
ladies  to  each  gentleman.  How  many  were  there  of  each  at 
first? 


118  SIMPLE     EQUATIONS. 

7.  Find  two  numbers,  such  that  the  sum  of  five  times  the 
first  and  twice  the  second  is  19;  and  the  difference  between 
seven  times  the  first  and  six  times  the  second  is  9. 

8.  Two  opposing  armies  number  together  21,110  men; 
and  twice  the  number  of  the  greater  army  added  to  three 
times  that  of  the  less  is  52,219.  How  many  men  in  each 
army? 

9.  A  certain  number  is  expressed  by  two  digits.  The 
sum  of  these  digits  is  11 ;  and  if  13  be  added  to  the  first 
digit,  the  sum  will  be  three  times  the  second.  What  is  the 
number  ? 

10.  A  and  B  possess  together  I570.  If  A's  share  were 
three  times  and  B's  five  times  as  great  as  each  really  is,  then 
both  would  have  I2350.     How  much  has  each  ? 

11.  If  I  be  added  to  the  numerator  of  a  fraction,  its  value 
is  I ;  and  if  i  be  added  to  the  denominator,  its  value  is  ^. 
What  is  the  fraction. 

12.  A  owes  $1200;  B,  $2550.  But  neither  has  enough  to 
pay  his  debts.  Said  A  to  B,  Lend  me  |  of  your  money, 
and  I  shall  be  enabled  to  pay  my  debts.  B  answered,  I 
can  discharge  my  debts,  if  j^^ou  lend  me  J  of  yours.  What 
sum  has  each  ? 

13.  Find  two  numbers  whose  difference  is  14,  and  whose 
sum  is  48. 

14.  A  house  and  garden  cost  I8500,  and  the  price  of  the 
garden  is  ^  the  price  of  the  house.     Find  the  price  of  each. 

15.  Divide  50  into  two  such  parts  that  f  of  one  part, 
added  to  f  of  the  other,  shall  be  40. 

16.  Divide  I1280  between  A  and  B,  so  that  seven  times 
A's  share  shall  equal  nine  times  B's  share. 

17.  The  ages  of  two  men  differ  by  10  years  ;  15  years  ago, 
the  elder  was  twice  as  old  as  the  younger.  Find  the  age  of 
each. 

18.  A  man  owns  two  horses  and  a  saddle.  If  the  saddle, 
worth  $50,  be  put  on  the  first  horse,  the  value  of  the  two 
is  double  that  of  the  second  liorse  ;  but  if  the  saddle  be  put 


TWO     UNKNOWN     QUANTITIES.  119 

on  the  second  horse,  the  value  of  the  two  is  $15  less  than 
that  of  the  first  horse.     Kequired  the  value  of  each  horse. 

19.  A  war-steamer  in  chase  of  a  ship  20  miles  distant, 
goes  8  miles  while  the  ship  sails  7.  How  far  will  each  go 
before  the  steamer  overtakes  the  ship  ? 

20.  There  are  two  numbers,  such  that  ^  the  greater  added 
to  -J  the  less  is  13 ;  and  if  I  the  less  be  taken  from  |  the 
greater,  the  remainder  is  nothing.     Find  the  numbers. 

21.  The  mast  of  a  ship  is  broken  in  a  gale.  One-third  of 
the  part  left,  added  to  ^  of  the  part  carried  away,  equals 
28  feet;  and  five  times  the  former  part  diminished  by 
6  times  the  latter  equals  12  feet.  What  was  the  height  of 
the  mast  ? 

22.  A  lady  writes  a  poem  of  half  as  many  verses  less  two 
as  she  is  years  old ;  and  if  to  the  number  of  her  years  that 
of  her  verses  be  added,  the  sum  is  43.  How  old  is  she  ? 
How  many  verses  in  the  poem  ? 

23.  What  numbers  are  those  whose  difference  is  20,  and 
the  quotient  of  the  greater  divided  by  the  less  is  3  ? 

24.  A  man  buys  oxen  at  $65  and  colts  at  $25  per  head, 
and  spends  $720  ;  if  he  had  bought  as  many  oxen  as  colts, 
and  vice  versa,  he  would  have  spent  1 1440.  How  many  of 
each  did  he  purchase  ? 

25.  There  is  a  certain  number,  to  the  sum  of  whose 
digits  if  you  add  7,  the  result  will  be  3  times  the  left-hand 
digit ;  and  if  from  the  number  itself  you  subtract  18,  the 
digits  will  be  inverted.    Find  the  number. 

26.  A  and  B  have  jointly  $9800.  A  invests  the  sixth  part 
of  his  property  in  business,  and  B  the  fifth  part  of  his,  and 
each  has  then  the  same  sum  remaining.  What  is  the  entire 
capital  of  each  ? 

27.  A  purse  holds  six  guineas  and  nineteen  silver  dollars. 
Now  five  guineas  and  four  dollars  fill  |-J  of  it.  How  many 
will  it  hold  of  each  ? 

28.  The  sum  of  two  numbers  is  a,  and  the  greater  is  n 
times  the  less.    What  are  the  numbers  ? 


120  SIMPLE     EQUATION". 


THREE    OR    MORE    UNKNOWN    QUANTITIES. 

225.  The  preceding  methods  of  elimination  of  two 
unknown  quantities  are  applicable  to  equations  containing 
three  or  more  unknown  quantities.     (Arts.  222-224.) 

226.  To  Solve  Equations  containing  three  or  more  Unknown 

Quantities. 

I.  Given  3a;  +  2^  —  52?  =  8,  2X  -{-  $y  -\-  4Z  =  16,  and 
$x  —  6y  ■}-  ^z  =  6,  to  find  x,  y,  and  z. 


-.UTlON.— By  the  problem. 

30!+  2y-  sz  = 

8 

(I) 

K                             ft 

2X+  3y+  4^  = 

16 

(2) 

tt                             U 

SX—  6y+  3Z  = 

6 

(3) 

Multiplying  (i)  by  2, 

6x+  4y—ioz  = 

16 

(4) 

(2)  by  3, 

6x+  gy+i2Z  = 

48 

(5) 

Subtracting  (4)  from  (5), 

5.?/  +  22S  = 

32 

(6) 

Multiplying  (2)  by  5, 

ioa;  + 1 5^  +  202  = 

80 

(7) 

(3)  by  2, 

1005— i2y+   6s  = 

12 

(8) 

Subtracting  (8)  from  (7), 

27y  +  i4Z  = 

68 

(9) 

Multiplying  (6)  by  27, 

1352^  +  5942  = 

864 

(10) 

(9)  by  5, 

1352/ +  70s  = 

340 

(II) 

Subtracting  (11)  from  (10), 

5242  = 

524 

T 

(12) 

Substituting  the  value  of  2  in  (6),                 y  = 

1 

2 

Substituting  the  value  of  y 

and  z  in  (2),      x  = 

3 

Ana.  x  =  3,  y  =  2,  2=1. 

Hence,  the 

Rule. — I.  From  thegiveti  equations  eliminate  one  unknown 
quantity,  by  combining  one  equation  with  another, 

II.  From  the  resulting  equations  eli?ninate  another  unknown 
quantity  in  a  similar  manner.  Continue  the  operation  ufitil 
a  single  equation  is  obtained,  with  but  one  unknown  quan- 
tity, and  reduce  this  by  the  preceding  rules. 

Note. — The  letter  having  the  smallest  coefiScients  should  be  elimi- 
nated first  ;  and  if  each  letter  is  not  found  in  all  the  given  equations, 
begin  witli  that  which  is  in  the  least  number  of  the  equations. 

226.  What  iB  the  rale  for  solving  equation?  having  three  or  more  unknown  quan- 
tities ?    Note.  Which  letter  should  be  eliminated  flret  ? 


THEEE     OR     MORE     UNKNOWN     QUANTITIES.  121 

Eeduce  the  following  equations : 

2.  5^  —  3«/  +  2^  =  28,  5.  5ic  +  2?/  4-  42;  =  46, 
SX  +  2y  —  4Z  =  15,  32;  +  2?/  +  z  =  23, 
3^  +  42;  —  a;  =  24.                  loa;  +  5?/  +  4^  =  75. 

3.  2a;  4-  5«/  —  32^  =  4,  6.  a;  +  ?/  +  2;  z=  53, 
4^  —  3y  +  22;  =  9,  a;  +  2y  4-  32;  =  105, 
5x  +  6y—2z=ziS.  ic  +  32^  +  40  =  134. 

4.  2a;  -\-sy  —  4Z  =  20,  7.     33:  +  42;  =  57, 

^—2?/ +  3^=    6,  2y—    z  =  ii, 

SX  —  2y-}-sz  =  26.  .    5a:  +  3?/  =:  65. 

234  345  450 

9.  Required  the  value  of  w,  x,  y,  and  z  in  the  following 

equations : 

w+x-\-y-\-z=i4  (i) 

2W  4-    a;  +    2/  —    2?  =    6  (2) 

2w  -\-  z^  —    y  +    z  =  14  (3) 

«^  —    ^  +  32/  +  4^  =  31  (4) 

SOLUTION. 

Adding  (i)  and  (2),  2)'^+  2X-\-   2y        =    20        (5) 

"       (2)  and  (3),  410+   4X  =20        (6) 

Multiply  (3)  by  4,  Sw+i2X—  4^+42=    56        (7) 

Subtract  (4)  from  (7),      7z^+i3a;—  7^        =    25        (8) 
Multiply  (5)  by  7,  2it^+i4a;  +  i4y        =140        (9) 

"         (8)  by  2,  14^^  +  26a!—  i4y        =    50      (10) 

Add  (9)  and  (10),  35?5  +  4ac  =190      (11) 

Multiply  (6)  by  10,         40^0  +  401?  =200      (12) 

Subtract  (11)  from  (12),  5«/J  =    10 

.'.    w  =      2 
Substituting  value  of  w  in  (6),  8  +  4aj  =    20 

.'.    «—      3 
Substituting  value  of  w  and  a?  in  (5),  etc., 
y  =  4,    and    2  =  5. 
6 


L;i2 


SIMPLE     EQUATIONS. 


227.  The  solution  of  equations  containing  many  unkno\;\'t] 
quantities  may  often  be  shortened  by  substituting  a  single 
letter  for  several. 


lo.  Eequired  the  value  of  w,  x,  y, 
and  z  in  the  adjoining  equations. 


io-^x-\-y=  13 
w-\-x-\-z  =  17 
iv-\-y-i-z  =  18 
x+y-{-z  =  2i 


(I) 
(2) 

IS) 
(4) 


Note. — Substituting  s  for  the  sum  of  the  four  quantities,  we  have, 

8  =  w  +  x+y  +  z. 

Equatien  (i)  contains  all  the  letters  but  z,  s—z  =  13  (5) 

(2)  «            «                "        p,  s-y  =  17  (6) 

(3)  "            "                '*        X,  s-x  =18  (7) 

(4)  "            "                "        w,  s-w  =  21  (8) 

Adding  the  last  four  equa- ) 


tions  together. 


or  4S—{z  +  y-{-x  +  w)  j^  =  69 
or  4s— 8 

That  is. 


3s  =  6g 

.-.      8=  23 

Substituting  23  for  s  in  each  of  the  four  equations,  we  have, 
w  =  2,    x=  s,    y  =  (>,    z=  10. 


II.  Eequired  the  value  of  v, 
Wf  X,  y,  and  z,  in  the  adjoining 
equations. 


(9) 
(10) 


V  -\-w-\-x-\-y  :=  10  (1) 

V  -\-W-\-X-\-Z  =  II  (2) 

V  -\-w-\-y-\-z=i2  (3? 

V  +x  -\-yi-z  =  is  (4) 

lw  +  x+y-{-z  =  i4  (5) 

Note. — Adding  these  equations,  4v  +  4w  +  4X  +  4y+4Z  =  60  (6) 

Dividing  (6)  by  4,                        v+  w+  x+  y+  z  =  15  (7; 
Subtracting  each  equation  from  (7),  we  have, 

s  =  5,    y  =  4,    x  =  3,    w  =  2,    and    v  =  1. 


I£.      W  +  X  -\-  Z  z=  10, 

X  -}-  y  +  z  =  12, 
w  +x  -\-y=  9, 
w  -\-  y  -\-  z  =  II. 


13- 


I       I 

X      y 


5 
6' 


y      z       12' 

X      z        4 


(See  Appendix,  p.  286.) 


THREE     OR  MORE     UNKNOWN     QUANTITIES.  123 


PROBLEMS. 

1.  A  man  has  3  sons ;  the  sum  of  the  ages  of  the  first 
and  second  is  27,  that  of  the  first  and  third  is  29,  and  of 
the  second  and  third  is  32.    What  is  the  age  of  each  ? 

2.  A  butcher  bought  of  one  man  7  calves  and  13  sheep 
for  $205  ;  of  a  second,  14  calves  and  5  lambs  for  I300;  and 
of  a  third,  12  sheep  and  20  lambs  for  $140,  at  the  same 
rates.     What  was  the  price  of  each  ? 

3.  The  sum  of  the  first  and  second  of  three  numbers  is 
13,  that  of  the  first  and  third  16,  ana  that  of  the  second 
and  third  19.     What  are  the  numbers  ? 

4.  In  three  battalions  there  are  1905  men:  J  the  first 
with  J  in  the  second,  is  60  less  than  in  the  third ;  |  of  the 
third  with  |  the  first,  is  165  less  than  the  second.  How 
many  are  in  each  ? 

5.  A  grocer  has  three  kinds  of  tea:  12  lbs.  of  the  firsts 
13  lbs.  of  the  second,  and  14  lbs.  of  the  third  are  together 
worth  $25  ;  10  lbs.  of  the  first,  17  lbs.  of  the  second,  and 
II  lbs.  of  the  third  are  together  worth  $24;  6  lbs.  of  the 
first,  12  lbs.  of  the  second,  and  6  lbs.  of  the  third  are  together 
worth  1 1 5.    What  is  the  value  of  a  pound  of  each  ? 

6.  Two  pipes,  A  and  B,  will  fill  a  cistern  in  70  minutes, 
A  and  0  will  fill  it  in  84  minutes,  and  B  and  C  in  140  min. 
How  long  will  it  take  each  to  fill  the  cistern  ? 

7.  Divide  $90  into  4  such  parts,  that  the  first  increased 
by  2,  the  second  diminished  by  2,  the  third  multiplied  by  2, 
and  the  fourth  divided  by  2,  shall  all  be  equal. 

8.  The  sum  of  the  distances  which  A,  B,  and  C  have 
traveled  is  62  miles;  As  distance  is  equal  to  4  times  C's, 
added  to  twice  B's;  and  twice  A's  added  to  3  times  B's,  is 
equal  to  17  times  C's.     What  are  the  respective  distances  ? 

9.  A,  B,  and  C  purchase  a  horse  for  $100.  The  payment 
would  require  the  whole  of  A's  money,  with  half  of  B's ;  or 
the  whole  of  B's  with  i  of  C's;  or  the  whole  of  C's  with  J 
of  A's.     How  much  money  has  each  ? 


OHAPTEE    XI. 
GENERALIZATION. 

228.  Generalisation  is  the  process  of  finding  a 
formula,  or  general  rule,  by  which  all  the  problems  x)f  a 
class  may  be  solved. 

229.  A  Problem  is  generalized  when  stated  in 
general  terms  which  embrace  all  examples  of  its  class. 

230.  In  all  General  I^roMems  the  quantities  are 
expressed  by  letters, 

I.  A  marketman  has  75  turkeys;  if  his  turkeys  are  mul- 
tiplied by  the  number  of  his  chickens,  the  result  is  225. 
How  many  chickens  has  he  ? 

Note. — This  problem  maybe  stated  in  the  following  general  terms : 

231.  The  Product  of  two  Factors  and  one  of  the  Factors 

being  given,  to  Find  the  other  Factor.* 

SuGGESTiON.^If  &  product  of  two  factors  is  divided  by  one  of  them, 
it  is  evident  the  quotient  must  be  the  other  factor.  Hence,  substituting 
a  for  the  product,  h  for  the  given  factor,  we  have  the  following 

General  Solution. — ^Let  x  =  the  required  factor. 

By  the  conditions,        xxb,OThx  =  a,the  product.     Hence,  the 

FOKMULA.  X  =  ^' 

h 

Translating  this  Formula  into  common  language,  we  have  the 
following 

EuLE. — Divide  the  product  ly  the  given  factor  ;  the  quo- 
tient is  the  factor  required. 

228.  What  is  generalization  ?  129.  When  is  a  problem  generalized  ?  230.  Hov» 
are  quantities  expressed  in  general  problems  ? 

*  New  Practical  Arithmetic,  Article  93. 


GENERALIZATIOIS'.  125 

Generalize  the  next  two  problems : 

2.  A  rectangular  field  contains  480  square  rods,  and  the 
length  of  one  side  is  16  rods.  What  is  the  length  of  the 
other  side  ? 

3.  Divide  576  into  two  such  factors  that  one  shall  be  48. 

4.  The  product  of  A,  B,  and  C's  ages  is  61,320  years;  A 
Is  30  years,  B  40.     What  is  the  age  of  C  ? 

Note. — The  items  here  given  may  be  generalized  as  follows  : 

232.  The  Product  of  three  Factors  and  two  of  them  being 
given,  to  Find  the  other  Factor. 

Suggestion.  —Substituting  a  for  the  product,  h  for  one  factor,  and 
e  for  the  other,  we  have  the 

General  Solution. — Let  x  =  the  required  factor. 

By  the  conditions,      xxbxc,    or  bcx  =  a,  the  product. 

Removing  the  coeflBcient,  we  have  the 

FOKMULA.  X  =  i — 

be 

Rule. — Divide  the  given  product  hy  the  product  of  the 
given  factors  ;  the  quotient  is  the  required  factor, 

5.  The  contents  of  a  rectangular  block  of  marble  are  504 
cubic  feet ;  its  length  is  9  feet,  and  its  breadth  8  feet.  What 
is  its  height  ? 

6.  The  product  of  3  numbers  is  62,730,  and  two  of  its 
factors  are  41  and  45.     Required  the  other  factor. 

7.  The  amount  paid  for  two  horses  was  $392,  and  the 
difference  in  their  prices  was  1 1 8.  What  was  the  price  of 
each  ? 

Note. — From  the  items  given,  this  problem  may  be  generalized 
as  follows : 


231.  When  the  product  of  two  factors  and  one  of  the  factors  are  given,  how  find 
the  other  factor  ?  232.  When  the  product  of  three  fiactors  and  two  of  them  are 
given,  how  And  the  other  foctor  ? 


126  ge:n^eralization". 

233.  The  Sum  and  Difference  of  two  Quantities  being  given, 
to  Find  the  Quantities. 

Suggestion. — Since  the  sum  of  two  quantities  equals  the  greater 
filns  the  less  ;  and  the  less  plus  the  difference  equals  the  greater ; 
it  follows  that  the  sum  plus  the  difference  equals  twice  the  greater. 
Substituting  .s  for  the  sum,  d  for  the  difference,  g  for  the  greater, 
9xid  I  for  the  less,  we  have  the  following 

General  Solution. — Let  ^  =  the  greater  number, 

and  1=    '*  less             " 

Adding,  gr-f-?  =  «,  the  sum. 

Subtracting,  g—l  —  d,  the  difference. 

Adding  sum  and  difference,  2g  =  s+d 

8A-d 

Removing  coefficient,  g  = ,  greater. 

Subtracting  difference  from  sum,  2I  =  s—d 

g ^ 

Removing  coefficient,  I  =  — — ,  less.    Hence,  the 


Formulas. 

This  problem  may  be  solved  by  one  unknown  quantity. 


7  —  ^  —  ^ 

2 


FORMATION    OF    RULES. 

234.  Many  of  the  more  important  rules  of  Arithmetic 
are  formed  by  translating  Algehraic  Formulas  into  common 
language.  Thus,  from  the  translation  of  the  two  preceding 
formulas  into  common  language,  we  have,  for  all  problems 
of  this  class,  the  following  general 

Rule. — I.  To  find  the  greater,  add  half  the  sum  to  half 
the  difference. 

II.  To  find  the  less,  suUract  half  the  difference  from  half 
the  sum. 

8.  Divide  I1575  between  A  and  B  in  such  a  manner  that 
A  may  have  $347  more  than  B.     What  will  each  receive  ? 

233.  When  the  stun  and  difference  of  two  quantities  are  given,  how  find  the 
quantities  ?    234.  Give  the  rule  derived  from  the  last  two  formulas. 


GENERALIZATION".  137 

9.  At  an  election  there  were  2150  vot^s  cast  for  two 
persons  ;  the  majority  of  the  successful  candidate  was  346. 
How  many  votes  did  each  receive  ? 

10.  If  B  can  do  a  piece  of  work  in  8  days,  and  0  m  12 
days,  how  long  will  it  take  both  to  do  it  ? 

Note. — Regarding  the  work  to  be  done  as  a  unit  or  i,  the  problem 
may  be  thus  generalized : 

235.  The  Time  being  given  in  which  each  of  two  Forces  can 
produce  a  given  Result,  to  Find  the  Time  required  by  the  united 
Forces  to  produce  it. 

Suggestion. — Since  B  can  do  the  work  in  8  days,  he  can  do  i  eighth 
of  it  in  I  day,  and  C  can  do  i  twelfth  of  it  in  i  day.  Substituting  a 
for  8  days,  and  h  for  12  days,  we  have  the 

General  Solution. — Let  x  =  the  time  required. 

Dividing  x  by  a,  we  have  -  =  part  done  by  B, 

a 

X 

**        X  by  6,  we  have,  h~    **         **       ^' 

By  Axiom  q,  -  +  -  =  i,  the  work  done. 

a      J) 

Clearing  of  fractions,  bx+ax  =  db 

Uniting  the  terms,  (a  +  b)x  =  ab,B  and  C*s  time. 

Removing  tlie  coefficient,  we  have  the 

Formula.        x  = i-- 

HvLiE.— Divide  the  product  of  the  numbers  denoting  the 
time  required  ly  each  force,  hy  the  su7n  of  these  numbers ; 
the  quotient  is  the  time  required  by  the  united  forces. 

11.  A  cistern  has  two  pipes;  the  first  will  fill  it  in 
9  hours,  the  second  in  15  hours.  In  what  time  will  both 
fill  it,  running  together  ? 


235.  The  time  beiug  given  in  which  two  or  more  forces  can  produce  a  result,  how 
find  the  time  reauired  for  the  united  forces  to  produce  it  ? 


128  GEN^EKALIZ  ATIOl!f. 

1 2.  A  can  plant  a  field  in  40  hours,  and  B  can  plant  it  in 
50  hours.  How  long  will  it  take  both  to  plant  it,  if  they 
work  together  ? 

236.  In  Generalizing  Problems  relating  to  Per- 
centage, there  is  an  advantage  in  representing  the  quantities, 
whether  known  or  unknown,  by  the  initials  of  the  elements 
or  factors  which  enter  into  the  calculations. 

Note. — The  elements  0Yfact(y,'s  in  percentage  are, 
ist.  The  Base,  or  number  on  wliicli  percentage  is  calculated. 
2d.   The  Rate  per  cent,  which  shows  how  many  hundredths  of 
the  base  are  taken. 

3d.  The  Per^centage,  or  portion  of  the  base  indicated  by  the  rate. 
4th.  The  Amount,  or  the  hsiseplus  or  minus  the  percentage.  Thus, 
Let  b  =  the  base.  p  =  the  percentage. 

r  =  the  rate  per  cent,        a  =  the  amount. 

13.  A  man  bought  a  lot  of  goods  for  $748,  and  sold  them 
at  9  per  cent  above  cost.    How  much  did  he  make  ? 

Note. — The  items  in  this  problem  may  be  generalized  as  follows : 

237.  The    Base   and    Rate    being    given,    to    Find   the 

JPerceutage,* 

Suggestion. — Per  cent  signifies  hundredths;  hence,  any  given 
per  cent  of  a  quantity  denotes  so  many  hundredths  of  that  quantity. 
But  finding  a  fractional  part  is  the  same  as  multiplying  the  quantity 
by  the  given  fraction.  Substituting  h  for  the  cost  or  base,  and  /•  for 
the  number  denoting  the  rate  per  cent,  we  have  the 

General  Solution. — Let  p  =  the  percentage. 

Multiplying  the  base  by  the  rate,  br  =  p.    Hence,  the 

FoKMUiA.        p  =  hr. 

Rule. — Multiply  the  base  hy  the  rate  per  cent ;  the  product 
is  the  percentage.    Hence, 

Note. — Percentage  is  a  product,  the  factors  of  which  are  the  base 
and  rate. 


236.  Note.  What  are  the  elements  or  factors  in  percentage?    237.  When  base 
and  rate  are  given,  how  find  the  percentage  ? 

♦  New  Practical  Arithmetic,  Arts.  336 — 340. 


GENERALIZATION^.  129 

14.  The  population  of  a  certain  city  in  1870  was  45,385  ; 
in  1875  it  was  found  to  have  increased  20  per  cent.  What 
was  the  percentage  of  increase  ? 

15.  Find  37^  per  cent  on  $2763. 

16.  A  western  farmer  raised  1587  bushels  of  wheat,  and 
sold  37  per  cent  of  it.     How  many  bushels  did  he  sell  ? 

17.  A  teacher's  salary  of  $2700  a  year  was  increased  $336. 
What  per  cent  was  the  increase  ? 

Note. — The  data  of  this  problem  may  be  generalized  as  follows  : 

238.  The  Base  and  Percentage  being  given,  to  Find  the  Rate, 

Suggestion. — Percentage,  we  have  seen,  is  a  product,  and  the  hase 
is  one  of  its  factors  (Art.  237,  note) ;  tlierefore,  we  have  the  product 
and  one  factor  given,  to  find  the  other  factor,  (Art.  231.)  Substituting 
2>  for  the  product  or  percentage,  and  b  for  the  salary  or  haae,  we 
have  the 

General  Solution.— Let  r  =  the  required  rate. 
Then  (Art.  231),  p-r-h  =  r.    Hence,  the 

Formula.       r  =  ^' 
o 

Rule. — Divide  the  percentage  ly  the  lase;  the  quotient 
is  the  rate. 

18.  From  a  hogshead  of  molasses,  25.2  gallons  leaked  out. 
What  per  cent  was  the  leakage  ? 

19.  A  steamship  having  485  passengers  was  wrecked,  and 
291  of  them  lost.     What  per  cent  were  lost? 

20.  A  man  gained  $750  by  a  speculation,  which  was* 
25  per  cent,  of  the  money  invested.  What  sum  did  hb 
invest  ? 

Note. — The  particular  statement  of  this  problem  may  be  tran». 
formed  into  the  following  general  proposition  : 

238.  When  the  hase  and  percentage  are  given,  how  find  the  rate? 


130  GENERALIZ  ATIOIT. 

239.  The  PeVcentage  and  Rate  being  given,  to  Find  the  Base, 

Suggestion. — We  have  the  product  and  one  of  its  factors  given, 
to  find  the  otJier  factor.  (Art.  237,  note.)  Substituting  p  for  the 
percentage,  and  r  for  the  rate  per  cent  gained,  we  have  the 

General  Solution. — Let  6  =  the  base. 

Then  (Art.  237),    _p-r-7'  =  6.     Hence,  the 

Formula.        b  =  —' 
r 

KuLE. — Divide  the  percentage  by  the  rate,  and  the  quotient 

is  the  base. 

21.  A  paid  a  tax  of  I750,  which  was  2  per  cent  of  his 
property.    How  much  was  he  worth  ? 

22.  A  merchant  saves  8  per  cent  of  his  net  income,  and 
lays  up  $2500  a  year.     What  is  his  income  ? 

23.  xit  the  commencement  of  business,  B  and  C  were 
each  worth  $2500.  The  first  year  B  added  8  per  cent  to 
his  capital,  and  C  lost  8  per  cent  of  his.  What  amount 
was  each  then  worth  ? 

Note. — The  items  of  this  problem  may  be  generalized  thus : 

240.  The  Base  and  Rate  being  given,  to  Find  the  Amount, 

Suggestion. — Since  B  laid  up  8  per  cent.,  he  was  worth  his  origi- 
nal stock  plus  8  per  cent  of  it.  But  his  stock  was  100  per  cent,  or  once 
itself;  and  100  per  cent,  +  .08  =  108  per  cent  or  1.08  times  his  stock. 

Again,  since  C  lost  8  per  cent,  he  was  worth  his  original  stock 
minus  8  per  cent  of  it.  Now  100  per  cent  minus  8  per  cent  equals 
100  per  cent  —  .08  —  92  per  cent,  or  .92  times  his  capital.  Substituting 
b  for  the  base  or  capital  of  each,  and  r  for  the  number  denoting  the 
rate  per  cent  of  the  gain  or  loss,  we  have  the 

General  Solution.— Let         a  =  the  amount. 

Then  will  b(i+r)  =  a,  B's  amount. 

And  b  (i—r)  =  a,  C's  amount. 

Combining  these  two  results,  we  have  the 

Formula.        a  =  b{i  ±  /•)• 
KuLE. — Multiply  the  base  by  i  i  the  rate,  as  the  case  may 
require,  and  the  result  ivill  be  the  amount. 

239.  When  percentage  and  rate  are  given,  bow  find  the  base  ?  240,  How  find  the 
amount  when  the  base  and  rate  are  given  ? 


GENEKALIZATION.  13i 

Note. — When,  from  the  nature  of  the  problem,  the  amount  is  to  he 
greater  than  the  base,  the  multiplier  is  i  plus  the  rate ;  when  less,  the 
multiplier  is  i  .ninus  the  rate. 

24.  A  man  bought  a  flock  of  sheep  for  $4500,  and  sold  it 
25  per  cent  above  the  cost.     What  amount  did  he  get  for  it  ? 

25.  A  man  owned  2750  acres  of  land,  and  sold  33  per 
cent  of  it.     What  amount  did  he  have  left  ? 

241.  The  elements  or  factors  which  enter  into  computa- 
tions of  interest  are  the  principal^  rate,  time,  interest,  and 
amount.     Thus, 

Let  p  =  the  principal,  or  money  lent. 
**     r  =  the  interest  of  $1  for  i  year,  at  the  given  rate. 
**      t  =  the  time  in  years. 
"      i  =  the  interest,  or  the  percentage. 
"     a  ~  the  amount,  or  the  sum  of  principal  and  interest. 

26.  What  IS  the  interest  of  $465  for  2  years,  at  6  per  cent? 
Note, — The  data  of  this  problem  may  be  stated  in  the  following 

general  proposition  : 

242.  The  Principal,  the  Rate,  and  Time  being  given,  to  Find 

the  Interest, 

General  Solution.— Since  r  is  the  interest  of  $1  for  i  year, 
pxr  must  be  the  interest  of  p  dollars  for  i  year ;  therefore,  pr  x  t 
must  be  the  interest  of  p  dollars  for  t  years.    Hence,  the 

Formula.        i  =  prt. 

EuLE. — Multiply  the  principal  hy  the  interest  of  %i  for 
the  given  time,  and  the  result  is  the  interest. 

27.  What  is  the  interest  of  I1586  for  i  yr.  and  6  m.,  at 
8  per  cent  ? 

28.  What  is  the  int.  of  $3580  for  5  years,  at  7  per  cent  ? 

29.  What  is  the  amt.  of  $364  for  3  years,  at  5  per  cent  ? 

Note.— This  problem  may  be  stated  in  the  following  general  terms '. 


Note.  When  the  amount  is  j^reater  or  less  than  the  base,  what  is  the  multiplier? 

241.  What  are  the  elements  or  factors  which  enter  into  computations  of  interest? 

242.  When  the  principal,  rate,  and  time  are  given,  how  find  the  interest  ? 


132  GEN^ERALIZATION. 

243.  The  Principal,  Rate,  and  Time  being  given,  to  Find  the 

Amount, 

General  Solution. — Reasoning  as  in  the  preceding  article, 

the  interest  =  prt. 
But  the  amount  is  the  sum  of  the  principal  and  interest. 
/.    p+prt  =  a.    Hence,  the 

FoBMULA.       a  =  p-{-  prt. 

Rule. — Add  the  interest  to  the  principal,  and  the  result  is 
the  amount. 

30.  Find  the  amount  of  $4375  for  2  years  and  6  months, 
at  8  per  cent. 

31.  Find  the  amt.  of  $2863.60  for  5  years,  at  7  per  cent. 

244.  The  delation  between  the  four  elements  in  the 

Formula,      a  =  p  ^  prt, 
is  such,  that  if  any  three  of  them  are  given,  the  fourth  may 
be  readily  found.     (Art.  243.) 

245.  The  Amount,  the  Rate,  and  Time  being  given,  to  Find 

the  Principal. 

Transposing  the  members  and  factoring,  we  have  the 

Formula.       p 


1  -\-  rt 

32.  What  principal  will  amount  to  I1500  in  2  years,  at 
6  per  cent  ? 

2,3'  What  sum  must  be  invested  at  7  per  cent  to  amount 
to  $300  in  5  years  ? 

246.   The  Amount,  the  Principal,  and  the  Rate  being  given, 
to  Find  the  Time. 

Transposing  2>  and  dividing  hy  pi',  (Art.  244),  we  have  the 

Formula.        t  = ^' 

pr 

34.  In  what  time  will  $3500,  at  6  per  cent,  yield  $525 

interest  ? 

243.  The  amount  f  244.  What  is  the  relation  between  the  four  elements  in  the 
preceding  formula.  245.  When  the  amount,  rate,  and  time  are  given,  state  the 
formula,    246.  When  the  amount,  principal,  and  rate*  are  given,  state  the  formula. 


GENERALIZATIOK. 


133 


247.  The  Hour  and  Minute  Hands  of  a  Clock  being  together 
at  I2!VI.,  to  Find  the  Time  of  their  Cottjuncfioii  between  any 
two  Subsequent  Hours. 

35.  The  hour  and  minute  hands  of  a  clock  are  exactly 
together  at  12  o'clock.  It  is  required  to  find  how  long 
before  they  will  be  together  again. 

Analysis. — The  distance  around  the  dial  of  a  clock  is  12  hour 
spaces.  When  the  hour-hand  arrives  at  I,  the  minute-hand  has  passed 
12  hour  spaces,  and  made  an  entire  circuit.  But  since  the  hour-hand 
has  moved  over  one  space,  the  minute-hand  has  gained  only  11  spaces. 
Now,  if  it  takes  the  minute-hand  *  hour  to  gain  1 1  spaces,  to  gain 
I  space  will  take  -^j  of  an  hour,  and  to  gain  12  spaces  it  will  take  12 
times  as  long,  and  12  times  -^j  hr.  =  ^f  hr.  =  Tj\  hour.    Or, 

Let  X  =  the  time  of  their  conjunction. 

Then        11  spaces    :     12  spaces    ::     i  hour     :    a;  hours. 

Multiplying  extremes,  etc.,     iia;  =  12 

Removing  coefficient,  x  =    i  jJy  hr.,  or  i  hr.  5y\  min. 

36.  When  will  the  hour  and 
minute  hand  be  in  conjunction 
next  after  3  o'clock  ? 

Suggestion. — Substituting  a  for  i^ 
hr.,  the  time  it  takes  the  minute-hand  to 
gain  12  spaces,  h  for  the  given  number 
of  hours  past  12  o'clock,  t  for  the  time  of 
conjunction,  we  have  the  following 

General  Solution,  a  x  h  =  t,  the 
time  required.     Hence,  the 

Formula.       t  =  ah. 

Rule. — Multiply  the  time  required  to  gain  12  spaces  by 
the  given  hour  past  1 2  o^dock  ;  the  product  will  de  the  time 
of  conjunction. 

37.  At  what  time  after  6  o'clock  will  the  hour  and  minute 
hand  be  in  conjunction  ? 

38.  At  what  time  between  9  and  10  o'clock  ? 

(See  Appendix,  p.  287.) 


247.  What  is  the  formula  for  finding  when  the  hands  of  a  clock  will  be  in 
conjunction  ?    Translate  this  into  a  rule. 


CHAPTER    XII. 
INVOLUTION  * 

248.  Involution  is  finding  a  power  of  a  quantity. 

249.  A  Power  is  the  product  of  two  or  more  equal 
factors. 

Thus,  3x3  =  9;  axaxa  =  a^',  9  and  a^  are  powers. 

250.  Powers  are  divided  into  differe^it  degrees ;  as  first, 
second,  third,  fourth,  etc.,  the  name  corresponding  with  the 
number  of  times  the  quantity  is  taken  as  2i  factor  to  produce 
the  power. 

251.  The  First  Power  is  the  quantity  itself. 

The  Second  Poiver  is  the  product  of  two  equal  factors, 
and  is  called  a  square. 

The  Third  Power  is  the  product  of  three  equal  factors, 
and  is  called  a  cuhe,  etc. 

Note. — The  quantity  caUed  the  first  'power  is,  strictly  speaking,  not 
a  power,  but  a  root.  Thus,  <2^  or  a,  is  not  the  product  of  any  two  equal 
factors,  but  is  a  quantity  or  root  from  which  its  powers  arise. 

252.  The  Index  or  Exponent  f  of  a  power  is  a,  figure 
or  letter  placed  at  the  right,  above  the  quantity.  Its  object 
is  to  show  hoto  many  times  the  quantity  is  taken  as  a  factor 
to  produce  the  power. 

Thus,  «'  =  a,   and  is  called  the  first  power. 

a'^  =  axa,  the  second  power,  or  square, 
a^  =  axaxa,  the  third  power,  or  cube, 
<35*  =  ax  ax  ax  a,  the  fourth  power,  etc. 

248.  What  is  involution?  249.  A  power?  250.  How  divided?  251.  The  first 
power  ?   Second  power  1  Third  ?    252.  What  is  the  index  or  exponent  ?  Its  object  ? 

*  Involution,  from  the  Latin  involvere,  to  roll  up. 
f  Index  (plural,  indices),  Latin  indicare,  to  indicate. 
Exponent,  from  the  Latin  exponere,  to  set  forth. 


IKVOLUTIOK.  135 

Notes. — i.  The  index  of  the  first  power  being  i,  is  commonly 
omitted. 

2.  The  expression  «*  ig  read  "  a  fourth,"  "the  fourth  power  of  a" 
or  " a  raised  to  the  fourth  power ;"  x^  is  read,  " x  nt\v"  or  " the 
wth  power  of  x." 

Read  the  following:  a\  d^,  x\  y^%  %^,  IT,  cC". 

253.  Powers  are  also  divided  into  direct  and  reciprocal, 

254.  Direct  Powers  are  those  which  arise  from  the 
continued  multiplication  of  a  quantity  into  itself. 

Thus,  the  continued  multiplication  of  a  into  itself  gives  the  series, 
a,    a^,    a',    «*,    «^    a^,    etc. 

255.  Reciprocal  Powers  are  those  which  arise  from 
the  continued  division  of  a  unit  by  the  direct  powers  of  that 
4uantity.     (Art.  55.) 

Thus,  the  continued  division  of  a  unit  by  the  direct  powers  of  a 
^ves  the  series, 


I       I        I        I         I        I 

a'    ~a^*    ^3'    "^'    ¥5'    ¥' 


256.  Reciprocal  Powers  are  commonly  denoted  by 
prefixing  the  sign  —  to  the  exponents  of  direct  powers  of 
the  same  degree. 

Thus,      -^a~\    4;  =  «-^    — ,  =  a-',     -^  =  a"^,  etc. 

257.  The  difference  in  the  notation  of  direct  and  recip- 
rocal powers  may  be  seen  from  the  following  series : 

(I.)    a\    a\    a\    a\    a\     i,    \,   -^,   ^,   -^„    ^,  etc. 

(2.)     a^,    aS    a%    a\    a\    a^,    a-\  a'"^,   a"^,   aS   a~^  etc. 

Note. — The  first  half  of  each  of  the  above  expressions  is  a  series 
of  direct  powers ;  the  loM  half,  a  series  of  reciprocal  powers. 

258.  Negative  Exponents  are  the  same  as  the 
exponents  of  direct  powers,  with  the  sign  —  prefixed  to 
them. 

Note.  The  index  i  ?  253.  How  else  are  powers  divided?  254.  Direct  powers? 
255.  Reciprocal?    256.  How  is  a  reciprocal  power  denoted ? 


136  INVOLUTION. 

N0TES.-~i.  This  notation  of  reciprocal  powers  is  derived  from  the 
continued  division  of  a  series  of  direct  powers  by  their  root;  that  is,  by' 
subtracting  i  from  the  successive  exponents.     (Art.  113.) 

2.  The  use  of  negative  exponents  in  expressing  reciprocal  powers 
avoids  fractions,  and  therefore  is  convenient  in  calculations. 

3.  Direct  poicers  are  often  called  positive,  and  reciprocal  powers, 
negative.  But  the  student  must  not  confound  the  quantities  whose 
exponents  have  the  sign  +  or  -,  with  those  whose  coeJicientshaiYe  the 
sign  +  or  — .  This  ambiguity  will  be  avoided,  by  applying  the  term 
direct,  to  powers  with,  positive  exponents,  and  reciprocal,  to  those  with 
negative  exponents. 

259.  The   Zero  I^oiver  of  a  quantity  is  one  whose 
exponent  is  o  ;  as,  a°  ;  read,  *^  the  zero  power  of  a." 
Every  quantity  with  the  index  o,  is  equal  to  a  unit  or  i. 

For,  —  r=  «"-»  =  ao  (Art.  113) ;  but  —  =  i  ;  hence,  ««  =  i. 


SIGNS     OF    POWERS. 

260.  When  a  quantity  is  positive,  all  its  poivers  are 
positive. 

Thus,  axa  =  a"^',  axaxa=:  a^^  etc. 

When  a  quantity  is  negative,  its  even  powers  are  positive, 
and  its  odd  powers  negative. 

Thus,    —a  X  —a  =  a^ ;    —a  x  —a  x  —a  =  —a^,  etc. 


FORMATION    OF    POWERS. 

261.  All  JPowers  of  a  quantity  may  be  formed  by 
multiplying  the  quantity  into  itself.     (Art.  249.) 

262.  To  Raise  a  Monomial  to  any  Required  Power. 

The  process  of  involving  a  quantity  which  consists  of 
several  factors  depends  upon  the  following 

259,  What  is  the  zero  power  ?    To  what  is  a  quantity  of  the  zero  power  equal  ? 
a6o.  Rule  for  the  signs  ? 


INVOLUTION".  137 


PRINCIPLES. 

1°.  The  power  of  the  jwoduct  of  two  or  more  factors  is 
equal  to  the  product  of  their  powers. 

2°.  The  product  is  the  same,  in  whatever  order  the  factors 
are  tahen.     (Art.  87,  Prin.  3.) 

1.  Given  3^^  to  be  raised  to  the  third  power. 

SOLUTION. 

{UhJ  =  3a¥  X  3«^  X  sah^  (Art.  261), 
or,       sxsxsxaxaxaxb^xb^xIP  (Prin.  2), 

.-.     (3ab^y  =  27«3&6,  Ans. 
Involving  each  of  tliese  factors  separately,  we  have,  (3)3  =  27 ; 
(af  =  a^ ;  and  {¥f  =  ¥  ;  and  27xa^x¥  =  2']a^¥,  Ans.    Hence,  the 

EuLE. — Eaise  the  coefficient  to  the  poiuer  required,  and 
multiply  the  index  of  each  letter  by  the  index  of  the  power, 
prefixing  the  proper  sign  to  the  result,     (Art.  90.) 

Notes.— T.  A  single  letter  is  involved  by  giving  it  the  index  of  the 
required  power. 

2.  A  quantity  which  is  already  a  power  is  involved  by  multiplying 
its  index  by  the  index  of  the  required  power. 

3.  The  learner  must  observe  the  distinction  between  an  index 
and  a  coefficient.  The  latter  is  simply  a  multiplier,  the  former  shows 
how  many  times  the  quantity  is  taken  as  a,  factor. 

4.  This  rule  is  applicable  both  to  positive  and  negative  exponents. 

2.  What  is  the  square  of  abc  ? 

3.  What  is  the  square  of  —  abc  ? 

4.  What  is  the  cube  of  xyz  ? 

5.  What  is  the  fifth  power  of  abc  ? 

6.  What  is  the  fourth  power  of  2x^y  ? 

7.  What  is  the  third  power  of  6a^^  ? 

8.  What  is  the  fourth  power  of  $aM^c? 

9.  What  is  the  sixth  power  of  2a^c^'i 

10.  What  is  the  eighth  power  of  abcd'i 

11.  What  is  the  nih  power  of  xyz  ? 

262.  How  raise  a  monoinial  to  any  power?  Note.  A  pingle  letter ?  A" quantity 
already  a  power  ?    Distinction  between  index  and  coefficient  ? 


138  INVOLUTION. 

12.  Find  the  fifth  power  of  {a  +  hy. 

13.  Find  the  second  power  of  {a  -h  hY", 

14.  Find  the  nth  power  of  {x  —  «/)*". 

15.  Find  the  7it\\  power  of  {x  +  yy. 

16.  Find  the  second  power  of  (a^  +  If). 

17.  Find  the  third  power  of  {aW¥). 

263.  To  Involve  a  Fraction  to  any  required  Power* 

18.  What  is  the  square  of  — ^? 

EuLE. — Raise  both  the  numerator  and  denominator  to  the 
required  power. 

'i(it^ 
10.  Find  the  cube  of  ^ — • 
2a 

20.  Find  the  fourth  power  of  ^ 

21.  Find  the  square  of  ^— gTn* 

2 

22.  Find  the  wth  power  of  -• 

23.  Find  the  wth  power  of  — ^  • 

<y 

264.  A  compound  quantity  consisting  of  two  or  more 
terms,  connected  by  +  or  — ,  is  involved  by  actual  multi- 
plication of  its  several  parts. 

24.  Find  the  square  of  3a  +  Z>2.        Ans.  ^a^-^-SaU^-^hK 

25.  What  is  the  square  of  «  +  5  +  c  ? 

Ans.  0?  +  2ab  +  2ac  +  5^4-  2bc  +  &; 

26.  What  is  the  cube  of  a;  +  2?/  +  2  ? 

265.  It  is  sometimes  sufficient  to  express  the  power  of  a 
compound  quantity  by  exponents. 

Thus,  the  square  of  a  +  &  =  («  +  &)2 ;  the  nth  power  of  a&  +  c  +  3«?^  - 

363.  How  involve  a  fraction?    264.  How  involve  a  compound  quantity  « 


INVOLUTION- 


139 


L 

— 

:i 

- 

' 

FORMATION    OF    BINOMIAL    SQUARES. 

266.  To  Find  the  Square  of  a  Binomial  in  the  Terms  of 
its  Parts. 

1.  Given  two  numbers,  3  and  2,  to  find  the  square  of 
their  sum  in  the  terms  of  its  parts. 

Illustration.— Let  the  shaded  part  of  the  diagram  represent  the 
square  of  3 ; — each  side  being  divided  into 
3  inches,  its  contents  are  equal  to  3  x  3,  or 
9  sq.  in. 

To  preserve  the  form  of  the  square,  it  is 
plain  equal  additions  must  be  made  to  two 
adjacent  sides ;  for,  if  made  on  one  side,  or  on 
opposite  sides,  the  figure  will  no  longer  be  a 
square. 

Since  5  is  2  more  than  3,  it  follows  that 
two  rows  of  3  squares  each  must  be  added  at 

the  top,  and  2  rows  on  one  of  the  adjacent  sides,  to  make  its  length 
and  breadth  each  equal  to  5.  Now  2'into  3  plus  2  into  3  are  12  squares, 
or  tvyice  the  product  of  the  two  parts  2  and  3. 

But  the  diagram  wants  two  times  2  small  squares,  to  fill  the  comer 
on  the  right,  and  2  times  2,  or  4,  is  the  square  of  the  second  part.  We 
have  then  9  (the  square  of  the  first  part),  12  (twice  the  product  of  the 
two  parts  3  and  2),  and  4  (the  square  of  the  second  part).     Therefore, 

(3  +  2)2  =  32  +  2  X  (3  X  2)  +  2^. 

2.  Required  the  square  oix-^y,     Ans.  x^-{-2  xxy-\-y\ 

Hence,  universally. 

The  square  of  the  sum  of  two  quantities  is  equal  to  the 
square  of  the  first,  plus  twice  their  product,  plus  the  square 
of  the  second. 

Note. — The  square  of  a  Unomial  always  has  three  terms,  and  con- 
sequently is  a  trinomial.     Hence, 

No  binomial  can  be  a  perfect  square.    (Art.  loi.) 

266.  To  what  is  the  square  of  the  sum  of  two  quantities  equal  ?  How  illustrate 
the  square  of  the  sum  of  two  quautities  in  the  terms  of  its  parts  ? 


140  BII^OMIAL    THEOREM. 

267.  All  Binomials  may  be  raised  to  any  required 
power  by  continued  multiplication.  But  when  the  expo- 
nent of  the  power  is  large,  the  operation  is  greatly  abridged 
by  means  of  the  Bi7iomial  Theorem.* 

268.  The  Binomial  Hieorem  is  a  general  formula 
by  which  any  power  of  a  binomial  may  be  found  without 
recourse  to  continued  multiplication. 

To  illustrate  this  tlieorem,  let  us  raise  the  binomials  «  +  &  and  a—h 
to  the  second,  third,  fourth,  and  fifth  powers,  by  continued  multipli- 
cations : 

{a  +  hf  =  a^  +  2(0)  +  ¥. 

(a  +  hf  =  a^  +  sa^b  +  saf^  +  ¥. 

{a  +  Vf  =  a*  +  ^a%  +  ta%'^  +  40*3  +  j^^ 

{a  +  yf  =  a^  +  5a*&  +  loaW  +  loarW  +  506*  +  }fi. 

(a  -  bf  =  a?  -  2ab  +  b^. 

{a  -  bf  ^  «3  _  3^2^  +  2>ab''  —  b\ 

(a  -  bf  =  a'^  -  4a^b  +  6a?b^  -  ^ab^  +  b\ 

{a  -  bf  -oJ>  -  sa^ft  +  \oaW  -  loa^&a  +  5^54  _  y,^ 

269.  Analyzing  these  operations,  the  learner  will  discover 
the  following  laws  which  govern  the  expansion  of  Unomials : 

1.  The  number  of  terms  in  any  power  is  one  more  than  the 
index  of  the  power. 

2.  The  i7idex  of  the  first  term  or  leading  letter  is  the 
index  of  the  required  power,  which  decreases  regularly  by  i 
through  the  other  terms. 

The  index  of  the  following  letter  begins  with  i  in  the 
second  term,  and  increases  by  i  through  the  other  terms. 

3.  The  sum  o/the  indices  is  the  same  in  each  term,  and  is 
equal  to  the  index  of  the  power. 

268.  What  is  the  Binomial  Theorem  ?  269.  What  is  the  law  respecting  the  num- 
ber of  terms  in  a  power  ?  The  indices  of  each  quantity  ?  The  sum  of  the  indice* 
iu  each  term  ? 

*  This  method  was  discovered  bj  Sir  Isaac  Newton,  in  1666. 


BINOMIAL     THEOEEM.  141 

4.  The  coefficient  oi  the  first  and  last  term  of  every  power 
is  I  ;  of  the  second  and  next  to  the  last,  it  is  the  tJidex  of 
the  power;  and,  universally,  the  coefficieuts  of  any  two 
terms  equidistant  from  the  extremes,  are  equal  to  each  other. 

Again,  the  coefficients  regularly  increase  in  the  first  half 
of  the  terms,  and  decrease  at  the  same  rate  in  the  last  half. 

5.  The  signs  follow  the  same  rule  as  in  multiplication 

270.  The  preceding  principles  may  be  summed  up  in  the 
following 

GENERAL    RULE. 

I.  In'DICES. — Give  the  first  term  or  leading  letter  the  index 
if  the  required  power,  and  diminish  it  regularly  hy  i  through 
the  other  terms. 

TJie  index  of  the  following  letter  in  the  second  term  is  i, 
and  increases  regularly  hy  i  through  the  other  terms. 

II.  Coefficients. — The  coefficient  of  the  first  term  is  i. 
To  the  second  term  give  the  index  of  the  poiuer ;  and, 

universally,  multiplying  the  coefficient  of  any  term  hy  the 
index  of  the  leading  letter  in  that  term,  and  dividing  the 
product  by  the  index  of  the  following  letter  increased  hy  i, 
the  result  will  he  the  coefficient  of  the  succeeding  term. 

III.  Signs. — If  hoth  terms  are  positive,  mahe  all  the  terms 
positive;  if  the  second  term  is  negative,  make  all  the  odd 
terms,  counting  from  the  left,  positive,  a?id  all  the  even  terms 


the  binomial  formula. 

n—-  I 
(a  +  hy  =  a''  -\-n  x  a""-^  h  +  n  x a^'-^h^   etc. 

Note. — The  preceding  rule  is  based  upon  the  supposition  that  the 
index  is  a  positive  whole  number  ;  but  it  is  equally  true  when  the  index 
is  e\ih.Qr  positi've  or  negative,  integral  or  fractional. 

The  coeflacients  of  the  first  and  last  terms  ?  The  law  of  the  signs  ?  270.  What 
is  the  general  rule  ? 


142  BINOMIAL     THEOKEM. 

Expand  the  following  binomials : 

1.  {a  +  by.  6.     (^-\-zy^. 

2.  (a  —  by.  7.     (a  —  by. 

3.   {c  +  ay,  8.   (m  +  7iy\ 

4.     {x  +  yy,  g.     (x—y)^. 

5-     {^—yy*  lo.     {a+by. 

2T1..  When  the  terms  of  a  binomial  have  coefficients  or 
exponents,  the  operation  may  be  shortened  by  substituting 
for  them  single  letters  of  the  first  power.  After  the  opera- 
tion is  completed,  the  value  of  the  terms  must  be  restored. 

11.  Required  the  fifth  power  of  i^  +  3^2 
Solution.— Substitute  a  for  x'^,  and  h  for  32^2 .  then 

(a  +  If  =  a^  +  5a^&  +  loaW  +  ioa?l^  +  soft* + 6^. 
Restoring  the  values  of  a  and  &, 

(a;2  +  3^2)5  _  ajio  +  1 53,8^2  ^  goa?^^  +  2-joxh^  +  4052;^  +  2432/^®. 

12.  Expand  (x^  —  2,by. 

A71S.  7^  —  i2^b  +  s^b^  —  io8ic2^»3  4.  8iJ4. 

272.  Every  power  of  i  is  i,  and  when  a  factor  it  has  no 
effect  upon  the  quantity  with  whicli  it  is  connected.  (Art.  94, 
note.)  Hence,  when  one  of  the  terms  of  a  binomial  is  i,  it 
is  commonly  omitted  in  the  required  power,  except  in  the 
first  and  last  terms. 

Note. — In  finding  the  exponents  of  such  binomials,  it  is  only 
necessary  to  observe  that  the  mm  of  the  two  exponents  in  each  term 
is  equal  to  the  index  of  the  power. 

13.  Expand  (ic  +  i)^.      15.  Expand  (\  ~  ay. 

14.  Expand  \b  —  i)*.      16.  Expand  (i  -f  of, 

(See  Appendix,  p.  287.) 
271.  When  the  terms  have  coefficients  or  exponents,  how  proceed  ? 


POWERS     or     POLYNOMIALS.  143 

273.  A  Polynomial  may  be  raised  to  any  power  by  actual 
multiplication,  taking  the  given  quantity  as  a  factor  as  many 
times  as  indicated  by  the  exyonent  of  the  required  power. 
But  the  operation  may  often  be  shortened  by  reducing  the 
several  terms  to  two,  by  substitution,  arid  then  applying  the 
Binomial  Formula. 

17.  Kequired  the  cube  of  x  -^  y  -\-  z. 

Solution. — Substituting  a  for  (y  +  e),  we  have  aj  +  (y +2)  =  aj  +  a. 
By  formula,  {x-vaf  =  q^-\- yi?a  +  ym^  +  a^ 

Restoring  the  value  of  a, 


274.    To   Square   2i   Polynomial   without   Recourse  to 
Multiplication. 

18.  Eequired  the  square  oi  a  -\-  h  -{•  c. 

Solution. — By  actual  multiplication,  we  have, 

{a  +  h^-cf  =  a'^  +  2ab+2ac  +  ¥  +  2bc+c^. 
Or,  changing  the  order  of  terms, 

a'^  +  ¥  +  c^  +  2<zb  +  2ac+2bc. 
Or,  factoring,  we  have,  a'^  +  2aib  +  c)  +  ¥  +  2bc  +  cK 

19.  Eequired  the  square  ot  a  -{-  b  -\-  c  +  d. 

Solution. — By  actual  multiphcation,  we  have, 

a^ +  1^  +  d^  +  d^  +  2db  +  2ac  +  2ad  +  2bc  +  2bd  +  2cd. 
Or,  changing  the  order  of  the  terms,  and  factoring,  we  have, 
a?  +  2a(b+c-\-d)->r  ¥  4-  2&  (c  +  cZ)  -4-  c^  +  2cd  +  d^.    Hence,  the 

KuLE. — To  the  sum  of  the  squares  of  the  terms  add  twice 
the  product  of  each  pair  of  terms. 

Or,  To  the  square  of  each  term  add  twice  its  product  into 
the  sum  of  all  the  terms  which  folloio  it. 

20.  Eequired  the  square  oix  -\-  y  -{-  z. 

21.  Eequired  the  square  of  a  —  h  -{-  c. 

22.  Eequired  the  square  oi  a  -\-  x  -\-  y  ■\-  z. 

273.  How  may  a  polynomial  be  raised  to  any  required  power  ?  274.  What  is  the 
rule  foi-  squaring  a  polynomial  ?^     _  - 


a 


OFT..E     -^X 


144  ADDITION     OF     POWERS. 

275.  When  one  of  the  terms  of  a  binomial  is  2^  fraction, 
it  may  be  involved  by  actual  multiplication,  or  by  reducing 
the  mixed  quantity  to  an  improper  fraction,  and  then 
involving  the  fraction.    (Art.  171.) 

22t.  Kequired  the  square  of  x  -{•  ^\  and  x  —  J. 

«  +  i  a;  —  i 

a-  +  i  a!-  i 

+  ja;  +  i  -ia;  +  i 

aj'+    X  +  \  a^—    X  +  \ 

Or,  reduce  the  mixed  quantities  to  improper  fractions.    Thus, 

a;  +  -  =  -= — ;        and        x = ,    (Art.  171.) 

22  22^'' 


/2a!+i\2     4a;-+4a;+i  .        /2aj— iV 


4a;2— 4a;+i 


Expand  the  following  mixed  quantities : 

24.     (fl^  +  J)^.  26.     (—  f  +  2al)cf. 


25 


276.  Powers  are  added  and  subtracted  hke  other 
quantities.  (Arts.  67,  77.)  For,  the  same  powers  of  the 
same  letters  are  like  quantities;  while  powers  of  different 
letters  and  different  powers  of  the  same  letter  are  unlike 
quantities,  and  are  treated  accordingly.    (Arts.  43,  44,) 

28.  To  7^2  +  5  («  4-  J)3  _  6a;  +  3^  +  «' 

Add  —  30^3  +  4  (^  +  ^f  -f  4a;  4-  43^2  —  fl< 

Ans.       4(f  +  g  (a  -^  by  —  2X  +  $0^  -{-  43^ -^  a^  —  a^ 

29.  From        3^3  +  5  J2  —  4(^8  4.  ^^^a  _  ^5 

Take    —  40^^  +  3^  +  3^^  —  5^  +  a* 

Ans.         7^8  +  22>2  _  7^8  4-  5^:3  4.  4^2  _  ^^s  _  ^4 

275.  How  involve  a  binomial,  when  one  term  is  a  fraction  ?  276.  How  are  power* 
added  and  subtracted  ?    Why  ? 


DIVISION     OF     POWERS.  145 


MULTIPLICATION    OF    POWERS. 
277.  To  Multiply  Powers  of  the  Same  Moot. 

1.  What  is  the  product  of  2>tt'^b^  multiplied  by  aWi 

Solution. — Adding  the  exponents  of  each  letter,   we  have  yi^ 
and  6^    Now  3«^  x  &^  =  3a66%  Ans.     (Art.  94.) 

2.  Multiply  ^a^¥  by  a~^l)~\ 

Solution. — Adding  the  exponents  of  each  letter,  as  before,  we 
have  3a'^62,  Ans.    Hence,  the 

EuLE. — Add  the  exponents  of  the  given  quantities,  and  the 
result  will  be  the  product.    (Art.  94.) 

Notes. — i.  This  rule  is  applicable  to  positive  and  negative  exponents. 

2.  Powers  of  different  roots  are  multiplied  hy  writing  them  one 
after  another. 

Multiply  the  following  powers: 

3.  fl^^  by  a^.  7.  a~*b  by  a~^bK 

4.  x~^  by  ar\  8.  a~*cd  by  a'^c^d^. 

5.  b-^  hj  bK  9.  ¥  c-^  y-^  hj  b-^  (^y\ 

6.  a"*  by  a\  10.  a^y^z^  by  a~^y-^^. 


DIVISION    OF    POWERS. 
278.  To  Divide  Powers  of  the  Same  Root, 

II.  Divide  a^  by  a\ 

Solution. — Subtracting  one  exponent  from  the  other,  we  have 
a^-T-a^  =  a?,  the  quotient  sought.    (Art.  113.)    Hence,  the 

Rule. — Subtract  the  exponent  of  the  divisor  from  that  of 
the  dividend  ;  the  result  is  the  quotient.     (Art.  113.) 

Note. — This  rule  is  applicable  to  positive  and  negative  exponents. 

••"•t.  How  multiply  powers  of  the  same  root  ?  Note.  Of  different  roots  1  278.  What 
is  tne  rule  for  dividing  powers  of  the  same  root  ? 


146  TRAKSFEREING     FACTORS. 

Divide  the  following  powers : 

12.  a^  by  ar\  i6.  oc^yz^  by  ar^y^z~\ 

13.  x~^  by  ic^.  17.  i2aPh~^c  by  3flf2Z'"^c^ 

14.  ¥  by  52.  18.  6a;4^V  ^y  22^-2  ?/;22. 

15.  c~^  by  c~s.  19.  6oa%^(^  by  5«~2Z>%-A 

279.  The  Method  of  denoting  Reciprocal  Powers  shows 
that  any  factor  may  be  transferred  from  the  numerator  of  a 
fraction  to  the  denominator,  and  vice  versa,  by  changing  the 
5i^;i  of  its  exponent  from  +  to  — ,  or  —  to  +.     (Art.  256.) 

20.  Transfer  the  denominator  of  —3  to  the  numerator. 

q6  I 

Solution.    ^  =  «^  ^  "T^ia  ~  ^^  ^  ^~^  =  a^a;-^,  Ans. 

21.  Transfer  the  denominator  of  --^  to  the  numerator. 

x~^ 

Solution.     — -=—  =  61^-^ — -  =  axaj^  =  aofi,  Ans. 

■^ 
»  ^5 

22.  Transfer  the  numerator  of —  to  the  denominator. 

01  a^      I         .       I       I        I  .         I        . 

Solution.     — =-  x  a5__^___i.^-5_  — ^   ^;^5^ 

2^     2^  y     o^    y  or^y 

2^.  Transfer  the  numerator  of —  to  the  denominator. 

y 

Solution.     —  =  — ^  x  -  =  -^ ,   Ans. 
y       a^     y     a^y 

ax~^ 

24.  Transfer  ar'^  to  the  denominator  of • 

y 

25.  Transfer  y^  to  the  numerator  of  j—^' 

26.  Transfer  d~^  to  the  denominator  of  —q— 

27.  Transfer  af*  to  the  numerator  of  — -•    - 

279.  What  inference  may  be  drawn  from  the  method  of  denoting  rec;pri>4.Al 
powers  ?    How  transfer  a  factor  ? 


OHAPTEE    XIIL 

EVOLUTION  * 

280.  Evolution  is  finding  a  root  of  a  quantity.  It  is 
often  called  the  Extraction  of  roots. 

281.  A  Hoot  is  one  of  the  equal  factors  of  a  quantity. 

Notes. — i.  Powers  and  roois  are  correlative  terms.     If  one  quantity 
is  a  power  of  another,  tlie  latter  is  a  root  of  the  former. 
Thus,  a^  is  the  cube  of  a,  and  a  is  the  cube  root  of  a^. 

2.  The  learner  should  observe  the  following  distinctions  : 
ist.  By  involution  a  product  of  equal  factors  is  found. 
2d.  By  evolution  a  quantity  is  resolved  into  equal  factors.    It  is  the 
reverse  of  involution. 

3d.  By  division  a  quantity  is  resolved  into  two  factors. 
4th    By  subtraction  a  quantity  is  separated  into  two  parts. 

282.  Boots,  like  powers,  are  divided  into  degrees  ;  as,  the 
square,  or  second  root;  the  cube,  or  third  root;  the  fourth 
root,  etc. 

283.  The  Square  Hoot  is  one  of  the  two  equal  factors 
of  a  quantity. 

Thus,  5  X  5  =  25,  and  axa  =  a^ ;  therefore  5  is  the  cquare  root  of 
25,  and  a  the  square  root  of  a^. 

284.  The  Cube  Hoot  is  one  of  the  three  equal  factors 
of  a  quantity. 

Thus,  3  X  3  X  3  =  27,  and  axaxa  =  a^;  therefore,  3  js  the  cube 
root  of  27,  and  a  is  the  cube  root  of  a^. 

280,  What  is  evolution?  281.  A  root?  Note.  Of  what  is  evolution  the  reverse? 
283.  What  is  the  square  root  ?    284.  Cube  root  ? 

*  From  t^ie  Latin  evolvere,  to  unfold. 


148  EVOLUTION. 

285.  Eoots  are  denoted  in  two  ways : 

ist.  By  prefixing  the  radical  sign  ^  to  the  quantity.* 
2d.   By  placing  a  fractional  exponent  on  the  right  of  the 
quantity. 

Thus,  ^/a  and  a*  denote  the  square  root  of  a. 

y\/a  and  a^  denote  the  cube  root  of  a,  etc. 
Notes.— I.  The  figure  placed  over  the  radical  sign,  is  called  the 
Index  of  the  Root,  because  it  denotes  the  name  of  the  root. 
Thus,  ^/a,  and  \^a,  denote  the  square  and  cube  root  of  a. 

2.  In  expressing  the  square  root,  it  is  customary  to  use  simply  the 
radical  sign  ^  ,  the  2  being  understood. 

Thus,  the  expression  /Y/25  =  5,  is  read,  ''the  square  root  of  25  =  5." 

3.  The  method  of  expressing  roots  hj  fractional  exponents  is  derived 
irom.  the  manner  of  denoting  powers  by  integral  indices. 

Thus,  a^=axaxaxa;  hence,  if  a'^  is  divided  into  four  equal 
factors,  one  of  these  equal  factors  may  properly  be  expressed  by  a. 

286.  The  numerator  of  a  fractional  exponent  denotes  the 
power,  and  the  denominator  the  root. 

Thus,  a^  denotes  the  cube  root  of  the  first  power  of  a;  and  a*  denotes 
the  fourth  root  of  the  third  power  of  a,  or  the  third  power  of  the 
fourth  root,  etc. 

Read  the  following  expressions : 

1.  «i  4.    bk  7.  dh  10.  af. 

2.  «1  5.     c^.  8.  m"^.  II.  «»*. 

3.  a^,  6.     x^.  9.  n^.  12.  x§, 

13.  Write  the  third  root  of  the  fourth  power  of  a. 

14.  Write  the  fifth  power  of  the  fourth  root  of  x. 

15.  Write  the  eighth  root  of  the  twelfth  power  of  y. 

287.  A  JPerfect  Power  is  one  whose  exact  root  can 
be  found.    This  root  is  called  a  rational  quantity. 

285.  How  are  roots  denoted?  286.  What  docs  the  numerator  of  a  fractional 
exponent  denote  ?    The  denominator?    287.  What  is  a  perfect  power  ? 

*  From  the  Latin  radix,  a  root. 

The  si^n  /v/"  is  a  corruption  of  the  letter  r,  the  initial  of  ra4i^* 


EVOLUTION".  149 

288.  An  Imperfect  'Power  is  a  quantity  whose  exact 

root  cannot  be  found. 

289.  A  Surd  is  the  root  of  an  imperfect  power.  It  is 
often  called  an  irrational  quantity. 

Thus,  5  is  an  imperfect  power,  and  its  square  root,  2.23  +  ,  is  a  surd. 

Note. — All  roots  as  well  as  poicers  of  i,  are  i.  For,  a  root  is  a 
factor,  wliich  multiplied  into  itself  produces  a  power  ;  but  no  number 
except  I  multiplied  into  itself  can  produce  i.    (Art.  272.) 

Thus,  r,  i^  I",  and  y^i,  /y/i,  /y/i,  etc.,  are  all  equal. 

290.  Negative  Exponents  are  used  in  expressing  roots  as 
well  as  j902(;er5.     (Arts.  255,  257.) 

Thus,    — T  =  a-i  ;    -T  =  a-i  ;    -r  =  a-n  ^ 

a*  a*  ^t 

291.  The  value  of  a  quantity  is  not  altered  if  the  index 
of  the  power  or  root  is  exchanged  for  any  other  index  of 
the  same  value. 

Thus,  instead  of  x^,  we  may  employ  x^,  etc,    Hence, 

'i92.  A  fractional  exponent  maybe  expressed  in  decimals. 
'fhus,   a^  =  a^  =  a°*^ .    That  is,  the  square  root  of  a  is  equal  to  the 
fifth  power  of  the  tenth  root  of  a. 

Express  the  following  exponents  in  decimals : 

16.  Write  a^  in  decimals.  19.  Write  h'^  in  decimals. 


1  . 


17.  Write  a^  in  decimals.  20.  Write  x^  in  decimals. 

18.  Write  a^  in  decimals.  21.  Write  y^  in  decimals. 

22.  Express  a^  in  decimals.  Ans.  a^  =  ^0.333333+  ^ 

23.  Express  x^  in  decimals.  Ans.  x^  =  2^0.66666+  ^ 

24.  Express  y^  in  decimals.  Ans.  y^  =  y-^. 

«5.  Write  a^  in  decimals.  Ans.  a~s  =  a^'^. 

Note. — In  many  cases,  &  fractional:  exponent  can  only  be  expressed 
approximately  by  decimals. 

288.  An  imperfecl  power  ?    289.  A  rard  ?    290.  Are  negative  exponents  used  in 
expressing  roots  1    292.  How  are  fractional  exDonents  sometimes  expressed  ? 


150  EVOLUTION. 

293.  The  Signs  of  Hoots  are  gc  remed  by  the  following 

PRINCIPLES. 

1°,  An  odd  root  of  a  quantity  has  the  same  sign  as  the 
quantity. 

■   2°.  A71  even  root  of  a  positive  quantity  is  either  positive 
or  negative,  and  has  the  double  sigti,  ±. 

Thus,  the  square  of  +a\aa^,  and  the  square  of  —a  is  a^ ;  therefore 
the  square  root  of  a^  may  be  either  +a  or  —a\  that  is,  ^/a^  =  ±  a. 

3°.  The  root  of  the  product  of  several  factors  is  equal  to 
the  product  of  their  roots. 

Notes. — i.  The  ambiguity  of  an  even  root  is  removed,  when  it  is 
knqwn  whether  the  power  arises  from  a  positive  or  a  negative  quantity. 

2.  It  should  also  be  observed  that  the  two  square  roots  of  a  positive 
quantity  are  numerically  equal,  but  have  contrary  signs. 

294.  An  Even  Hoot  of  a  7iegative  quantity  cannot  be 
found.    It  is  therefore  said  to  be  impossible. 

Thus,  the  square  root  of  —a^  is  neither  +a  nor  —a.  For,  +ax  +a 
=  +a^ ;  and  —a  x  —a  =  +a^.    Hence, 

295.  An  even  root  of  a  negative  quantity  is  called  an 
Imaginary  Quantity. 

Thus,  \/— 4,  V'— a^  'V^— a^  are  imaginary  quantities. 

296.  To  Find  the  B^oot  of  a  Monomial. 

I.  What  is  the  square  root  of  a^? 

Analysis. — Since  a'^  —  axa,  it  follows  that  one  opeeation. 

of  the  equal  factors  of  «?  is  a;  therefore,  a  is  its  ^/a?  =  a 

square  root.    (Art.  283.) 

Again,  since  multiplying  the  index  of  a  quantity  by  a  number 
raises  the  quantity  to  a  corresponding  power,  it  follows  that  dividing 
the  index  by  the  same  number  resolves  the  quantity  into  a  correspond- 
ing root.  Thus,  dividing  the  index  of  a^  by  2,  we  have  a'  or  a,  which 
is  the  square  root  of  a^. 

293.  What  principles  eovem  the  sign?,  of  roots  ?  When  is  the  doable  sisu  used  ? 
Illustrate  this.  JVote.  When  is  the  ambiguity  removed  ?  294.  What  is  an  even  root 
of  a  negative  quantity  ?    Illustrate.    295.  What  is  it  called  ? 


EVOLUTION.  151 

2.  What  is  the  square  root  of  9^^^  ? 

Analysis. — Since  9  =  3x3,  the  index  of  operation. 

jj*  =  2  X  2,   and  the   index  of  &2  =  i  x  2,   it  ^/oaf^  :=  2t^^b 

follows  ^shat  the  square  root  of  9  is  3,  that  of 
a^  is  a-.;  and  that  of  ¥  is  6^  or  &.    Therefore,  ^^a?¥  =  ^a^h.   Hence,  the 

Rule. — Divide  the  index  of  each  letter  hy  the  index  of 
the  required  root;  to  the  result  prefix  the  root  of  the 
coefficient  with  the  proper  sign.     (Art.  293.) 

Note. — This  rule  is  based  upon  Principle  3.  If  a  quantity  is  an 
imperfect  power,  its  root  can  only  be  indicated. 

296,  a.  Tlie  root  of  a  Fraction  is  found  hy  extracting  the 
root  of  ^ach  of  its  terms. 

Find  the  required  roots  of  the  following  quantities: 

3.  Va^.  10.     'v/36«^Z>2. 


"s/a^  or  a.  11.     ^/^^y\ 

3  /- 


^/  AfXy.  12.  v64a% 

^2>a%\  13.  ^i^xy. 

V^^ahc,  14.  A/49^y. 

V^i6^.  15.  ^/r^. 


297.    To  Extract  the  Square  Moot  of  ihe  Square  of  a 
Binomial. 

I.  Required  tb€  square  root  of  a^  +  2a5  4-  l\ 

Analysis.  —  Arrange  the  terms  operation. 

according  to  the  powers  of  the  letter  O^  ■\-  2ab  +  Z>2  (  fl^  -f  d 

a ;  the  square  root  of  the  first  term  ^2 

is  a,  which  is  the  first  term  of  the  \_  'h\      h  A-  Jvi 

root.    Next,  subtracting  its  square  '       /       ■,        j^ 

from  the  given  quantity,  bring  down  2^0  +  (r 
the  remainder,  2ab  +  W. 

296.  How  find  the  root  of  a  monomial  ?    iVb/«.— Upon  what  principle  is  this  rulf 
based  ?    296  a.  How  find  the  root  of  a  fraction  ? 


152  EVOLUTION. 

Divide  the  ist  term  of  this  rem.  by  2a,  double  the  root  Oms  found,  the 
quotient  h  is  the  other  term  of  the  root.  Place  &  both  in  the  root  and  on 
the  right  of  the  divisor.  Finally,  multiply  the  divisor,  thus  increased, 
by  the  second  term  of  the  root,  and  subtracting  the  product  from 
2ab  +  h^,  there  is  no  remainder.     Therefore,  a  +  &  is  the  root  required. 

The  square  root  of  a:-—2db+l^  is  found  in  the  same  manner,  the 
terms  of  the  root  being  connected  by  the  sign  — .    Hence,  the 

HuLE. — Find  the  square  roots  of  the  first  and  third  ter^n^ 
and  connect  them  hy  the  sign  of  the  middle  term, 

2.  What  is  the  square  root  of  a^  -{-  4X  -\-  4? 

3.  What  is  the  square  root  of  «2  _  2^5  4-  i  ? 

4.  What  is  the  square  root  of  i  +  2X  -\-  a^? 

5.  What  is  the  square  root  oi  x^  -\-  ^x  -\-  ^  ? 

6.  What  is  the  square  root  of  «^  —  a  +  }  ? 

7.  What  is  the  square  root  of  x^  -{- ix  -] ? 

4 

298.  To  Extract  the  Square  Root  of  a  Polynomial. 

8.  Required  the  square  root  of  4«* — i2a^-\-s^^+6a-{-i, 

OPBRATION. 

40*  —  i2fl^3  +  5^2  _f-  6«  4-  I  (  2a^  —  $a  —  i 

4a* 


^2- 

3«)  - 

I2fl3 

+  9«2 

+  6a+i 

4^2 

-6a- 

-0 

-4«2 
-4«^ 

+  6«+  I 
+  6a  +  I 

Analysis. — The  square  root  of  the  first  term  is  2a',  which  is  the 
first  term  of  the  root.  Subtract  its  square  from  the  term  used 
md  bring  down  the  remaining  terms.  Divide  the  remainder  by 
louble  the  root  thus  found  ;  the  quotient  —3a  is  the  next  term  of  the 
root,  and  is  placed  both  in  the  root  and  on  the  right  of  the  partial 
divisor.  Multiply  the  divisor  thus  increased  by  the  term  last  placed 
in  the  root,  and  subtract  the  product  as  before. 

Next,  divide  the  remainder  by  twice  the  part  of  the  root  already 
found,  and  the  quotient  is  —i,  which  is  placed  both  in  the  root  and 
on  the  right  of  the  divisor. 

297.  How  extract  the  square  root  of  the  square  of  a  binomial? 


EVOLUTIOK.  153 

Finally,  multiply  the  divisor,  thus  increased,  by  the  term  last 
placed  in  the  root,  and  subtracting  the  product,  as  before,  there  is  no 
remainder.     Therefore,  the  required  root  is  2a"^—3«—i.     Hence,  the 

KuLE. — I.  A  rra?ige  the  terms  according  to  the  jjowers  of 
some  letter,  beginning  ivith  the  highest,  find  the  square  root 
of  the  first  term  for  the  first  term  of  the  root,  and  subtract 
its  square  from,  the  given  quantity. 

IL  Divide  the  first  term  of  the  ''remainder  by  double  the 
root  already  found,  and  place  the  quotient  both  i7i  the  root 
and  on  the  right  of  the  divisor, 

III.  Multiply  the  divisor  thus  increased  by  the  term  last 
placed  in  the  root,  and  subtract  the  product  from  the  last 
dividend.  If  there  is  a  remainder,  proceed  with  it  as  before^ 
till  the  root  of  all  the  quantities  is  found. 

Proof. — Multiply  the  root  by  itself,  as  in  arithmetic. 

Note. — This  rule  is  essentially  the  same  as  that  used  for  extracting 
the  square  root  of  numbers. 

Extract  the  square  root  of  the  following  quantities : 

9.     x^  +  2xy  +  y"^  -{-  2XZ  +  2yz  +  z\ 


10. 

a^  —  4ab  -\-  2a  -{-  4J2  _  4J  _}.  i. 

II. 

a^  +  ^a^  +  4^2  _  4^2  _  85  -f  4. 

12. 

I  —  4^2  _|_  4^4  4-  22;  —  4^2^;  +  x\ 

13. 

4a^  —  i6a^  +  24^2  _  i6a  +  4. 

14. 

a^-ab+  ibK 

X^                      1/2 

15- 

2-1-  — • 

299.  The  fourth  root  of  a  quantity  may  be  found  by 
extracting  the  square  root  twice  j  that  is,  by  extracting  the 
square  root  of  the  square  root. 

Thus,  y^iea*  =  4a^,  and  /^4a^  =  2a.  Therefore,  2a  is  the  fourth 
root  of  i6a\ 

Proof.    2ax2a  =  4a^ ;  4a^  x  40^  =  i6a\ 

The  eighth,  the  sixteenth,  etc.,  roots  may  be  found  in  like 
manner. 

298.  Of  a  polynomial  ?    299.  How  find  the  fourth  root  of  a  quantity  ?   The  eighth  ? 


CHAPTER    XIV. 
RADICAL    QUANTITIES. 

300.  A  Madical  is  the  root  of  a  quantity  indicated  by 
the  radical  sign  or  fractional  exponent. 

Notes.— I.  The  figures  or  letters  placed  before  radicals  are  coefficients. 

2.  In  the  following  investigations,  all  quantities  placed  under  tlie 
radical  sign,  or  having  a  fractional  exponent,  whether  perfect  or 
imperfect  powers,  are  treated  as  radicals,  unless  otherwise  mentioned. 

301.  The  Degree  of  a  radical  is  denoted  by  its  index, 
or  by  the  denominator  of  its  fractional  exponent.  (Arts. 
285,  286.) 

Thus,  '^ax,  a^,  and  {a  +  Vf,  are  radicals  of  the  same  degree. 

302.  Lihe  Madicals  are  those  which  express  the 
same  root  of  the  same  quantity.  Hence,  like  radicals  are 
like  quantities.     (Art.  43.) 

Thus,  s^y/a'^— &  and  3/y/«^— &,  etc.,  are  like  radicals. 

REDUCTION    OF    RADICALS. 

303.  Reduction  of  Madicals  is  changing  their 
form  without  altering  their  value. 

304.  The  Simjylcst  Form  of  radicals  is  that  which 
contains  no  factor  whose  indicated  root  can  be  extracted. 
Hence,  in  reducing  them  to  their  simplest  form,  all  exact 
poivers  of  the  same  name  a«  the  root  must  be  removed  from 
under  the  radical  sign. 

300.  What  is  a  radical  ?  301.  How  is  tlie  decree  of  a  radical  denoted?  302.  What 
are  like  radicals  ?  303.  Define  reduction  of  radicals.  304.  What  is  the  simplest  form 
}t  radicals  ? 


EEDUCTIOK     OF     RADICALS.  155 

CASE     I. 
305.  To  Reduce  a  Radical  to  its  Simplest  Form. 


I.  Reduce  ^/\Wx  to  its  simplest  form. 

Analysis. — By  inspection,  we  ofebation. 

perceive  that  tlie  given  radical  is  ^ \Wx  =  ^ ^0?  X  2X 

composed  of  two  factors,  <^a?  and  ^^       / — g  / — 

IX,  the  first  being  a  perfect  square  " 


and  the  second  a  surd.    (Art.  289.)         .'.   ^/ l^a^X  =  ^aV2X 
Removing  ga^  from  under  the  rad- 
ical sign  and  extracting  its  square  root,  we  have  3a,  which  prefixed 
to  the  other  factor  gives  s^y'z^,  the  simplest  form  required. 

2.  Reduce  4^/a^  —  a^x  to  its  simplest  form. 

Analysis.  —  Factoring  operation. 

the  radical  part,  wejiave  4^^.^^  =  4^a^x{a-x) 

the  two  factors,  \/a^  and  3,-—         », 

^a—x,  the  first  being  a  

perfect  cube,  and  the  sec-  .'.  ^^ a^—a^X  =  /^a\^a—X 

ond  a  surd.     Remove   a^ 

from  under  the  radical  sign,  and  its  root  is  a,  which  multiplied  by 
the  coeflBcient  4,  and  prefixed  to  the  radical  part,  gives  ^a^a—x,  the 
simplest  form.    Hence,  the 

Rule. — I.  Resolve  ilie  radical  into  two  factors,  one  of 
which  is  the  greatest  power  of  the  same  name  as  the  root. 

II.  Extract  the  root  of  this  power,  and  multiplying  it  ly 
the  coefficient,  prefix  the  result  to  the  other  factor,  with  the 
radical  sign  letween  them. 

Notes. — i.  This  rule  is  based  upon  the  principle  that  the  root  of 
the  product  of  two  or  more  factors  is  equal  to  the  product  of  their 
roots. 

2.  When  the  radicals  are  small,  the  greatest  exact  power  they 
contain  may  be  readily  found  by  inspection. 


3.  Reduce  3V5o«^^^  to  its  simplest  form.     Ans.  isav^x. 

4.  Reduce  6\/s4^l/  to  its  simplest  form.     Ans.  iSxV 2y. 

305.  Becite  the  rule.    Ifote.  Upon  what  based  ? 


156  EEDUCTIOK     OF     EADICALS. 

306.    To   Find    in    large    Radicals   the  Greatest   Power 
corresponding  to  the  indicated  Root. 

5.  Eeduce  ViSya  to  its  simplest  form. 

OPERATION. 

4  )  1872  ^1872  =  ^4x4x9  X  V13 

4  )  468  "     =  A/144  X  V13 

9)117(13  V1872  =  12V135  ^'ns. 

Analysis. — Divide  the  radical  by  the  smallest  power  of  the  same 
degree  that  is  a  factor  of  it ;  the  quotient  is  468.  Divide  this  quan- 
tity by  4  ;  the  second  quotient  is  117.  Th%  smallest  power  of  the  same 
degree  that  will  divide  117,  is  9.  The  quotient  is  13,  which  is  not 
divisible  by  any  power  of  the  same  degree.  The  product  of  the 
divisors,  4x4x9=  144,  is  the  greatest  square  of  the  given  radical. 
Extracting  the  square  root  of  144,  we  have  i2/y/i3,  the  simplest  form 
required.     Hence,  the 

'Rv'LE,— Divide  the  radical  by  the  smallest  power  of  the 
same  degree  ivhich  is  a  factor  of  the  given  radical. 

Divide  this  quotient  as  before;  and  thus  'proceed  till  a 
quotient  is  obtained  which  is  not  divisible  by  any  power  of 
the  same  degree.  The  continued  product  of  the  divisors  will 
be  the  greatest  poiver  required. 

Note. — This  rule  is  founded  on  the  principle  that  the  'product  of 
any  two  or  more  square  numbers  is  a  square,  the  product  of  any  two 
or  more  cuhic  numbers  is  a  cuhe,  etc. 

Thus,  2'  X  32  z=  36  =  62 ;  and  28x38  =  216  =  e^. 

Eeduce  the  following  radicals  to  their  simplest  form : 

\^54a^c. 

yVga^  —  2'ja^b, 
'\/64a^y. 
</8i^. 
V46Sa^. 

306.  How  find  the  greatest  power  corresponding  to  the  indicated  root,  in  large 
radicals  ?    Jffote.  On  what  principle  is  this  rule  founded  ? 


6. 

Va^. 

12. 

7- 

A/8a2^. 

13- 

8. 

2  Vgxy. 

14. 

9- 

3^^. 

15. 

10. 

SV^iSS- 

16. 

II. 

6\/252«2J. 

17. 

EEDUCTION     OF     RADICALS.  157 


CASE    II. 

307.   To   Reduce   a   Rational    Quantity  to  the  Form  of  a 
Radical. 

I.  Keduce  ^a^  to  the  form  of  the  cube  root. 

Analysis.— The  cube  root  of  a  quantity,  we  operation. 

have  seen,  is  one  of  its  three  equal  factors.  (3^^)^  =  2'ja^ 

(Art.  284.)    Now  3a''  raised  to  the  third  power  .     ^ „2 \/^2T^cfi 

is  27a*.    Therefore  3^^  =  ^i']a^.     Hence,  the 

Rule. — Raise  the  quantity  to  tlie  power  denoted  hy  the 
given  root,  and  to  the  result  prefix  the  corresponding  radical 
sign. 

Note. — The  coeflBcient  of  a  radical,  or  any  factor  of  it,  may  be 
placed  under  the  sign,  by  raising  it  to  the  corresponding  power,  and 
placing  it  as  a  factor  under  the  radical  sign. 

2.  Reduce  2a^  to  the  form  of  the  cube  root. 

3.  Reduce  (2a  +  d)  to  the  form  of  the  square  root. 

4.  Reduce  {a  —  2b)  to  the  form  of  the  square  root. 

5.  Place  the  coefficient  of  saVb  under  the  radical  sign. 

6.  Place  the  coefficient  of  2a^\^ab  under  the  radical  sign. 

7.  Reduce  2x^y^z^  to  the  form  of  the  fourth  root. 

8.  Reduce  ^abc  to  the  form  of  the  cube  root. 

9.  Reduce  ^{a  —  b)  to  the  form  of  the  cube  root. 
10.  Reduce  a^  to  the  form  of  the  cube  root. 

Note. — When  a  power  is  to  be  raised  to  the  form  of  a  required 
root,  it  is  not  the  given  letter  that  is  to  be  raised,  but  the  power  of  the 
letter.  * 

I I.  Reduce  ah^  to  the  form  of  the  fourth  root. 

12.  Reduce  a  —  b  to  the  form  of  the  square  root. 

13.  Reduce  a'^  to  the  form  of  the  nth  root. 

307.  How  reduce  a  rational  quantity  to  the  form  of  a  radical?  Note.  How  place 
a  coefficient  under  the  radical  sign  ?  Note.  How  raise  a  power  to  the  form  of  * 
required  root  ? 


158  REDUCTIOK     OF     EADICALS, 


CASE    III. 

308.  To  Reduce  Radicals  of  different  Degrees  to  others  of 

equal  Value,  having  a  Common  Index. 

1.  Eeduce  a^  and  b^  to  equivalent  radicals  of  a  common 
index. 

Analysis. — The    fractional  opebation. 

indices  i  and  i,  reduced  to  a  J  =  -^  and  i  = -rj 

common  denominator  become  ^.^  a^  =  a^  and  b^  =  ^t\ 

j\  and  f        But  af'  =  {a^)^,  4         ,  ,.  i  ,    ,  3         ,„,  1 

and    5^^  =  (63p.       (Art.    174.)        ^"  "=  ^^  ^^  ^^^  ^"  ^  ^^^^ 
Therefore    (a^)^'^    and    (ft^)"    are  the  radicals  required.    Hence,  the 

EuLE. — I.  Eeduce  the  indices  to  a  common  denominator. 

II.  Raise  each  quantity  to  the  power  expressed  hy  the 
numerator  of  the  new  index,  and  indicate  the  root  expressed 
ly  the  common  denominator.     (Art.  174.) 

Eeduce  the  following  radicals  to  a  common  index : 

2.  a^  and  (hc)^.  7.     V4S  and  \^2aK 

3.  3^  and  5^.  8.     a^  and  5«. 

4.  a^  and  6^.  9.     l^  and  c». 

5'     Vs^,  V^3,  and  v^.         10.     («  +  h)^  and  (a-  <^)i 
6.     '\/2x^  and  Vs^^.  11.     (^  — ^)^  and  (a?  4  «/)i 

CASE    IV. 

309.  To  Reduce  a  Quantity  to  any  Hequired  Index^ 

I.  Eeduce  a^  to  the  index  ^. 

Analysis. — Divide  the  index  |  byl;  operation. 

wo  have  |  or  ^.    Place  this  index  over  a;  l""~^"f^=i^XT^^I 

it  becomes  <z',  and  setting  the  required  i"  "^  1  ^^^  F  ^^  7 

index  over  this,  the  result,  (a*)^,  is  the  /,  (^v^j  -<4ws. 
answer.    Hence,  the 

30^.  5ow  reduce  radicals  to  a  coininon  iude?  ? 


ADDITION"     OF     EADICALS,  159 

Rule. — Divide  the  index  of  the  given  quantity  hy  the 
required  index,  and  'placing  the  quotie7it  over  the  quantity, 
set  the  required  index  over  the  whole. 

Note. — This  operation  is  the  same  as  resolving  the  original  index 
into  two  factors,  one  of  which  is  the  required  index.    (Art.  126,  note.) 

2.  Eeduce  a^  and  ifi  to  the  index  J. 

Solution.    J-^i  =  |xf  =  |,  tlie  first  index. 

|-i-i  =  f  X  f  =  f ,  the  second  index. 
Therefore,  (a^)^  and  (&")*  are  the  quantities  required. 

3.  Reduce  3^  and  43  to  the  common  index  ^, 

4.  Reduce  a^  and  i^  to  the  common  index  \, 

5.  Reduce  a^  and  h^  to  the  common  index  J. 

6.  Reduce  a^  and  ifi  to  the  common  index  -f. 

7.  Reduce  w^  and  5^  to  the  common  index  J-. 

ADDITION    OF    RADICALS. 
310.  To  Find  the  Sutn  of  two  or  more  Radicals. 

1.  Wliat  is  the  sum  of  3  V»  and  5  Va  ? 

Analysis. — Since  these  radicals  are  opekation. 

of  the  same  degree  and  have  the  same  ^^/a  -f-  S^^  =  SV^ 

radical  part,  they  are  like  quantities. 

(Art.  43.)    Therefore  their  coefficients  may  be  added  in  the  same 
manner  as  rational  quantities.    (Art.  67.) 

2.  What  is  the  sum  of  3^8  and  4^/18  ? 
Analysis.— These  radicals  operation.  ^ 

are  of  the  same  degree,  but  3^8    =  3V4  X  ^2  ■=.    6^/2 

the  radical  parts  are  unlike ;  /—x  r~  r~  /~ 

^,       .         XI  .    -,  4Vi8  =  4VQ  X  V2  =  I2V2 

therefore,    they    cannot    be  ~t  ^  y         »  "^ 

united  in  their  present  form.  /.   6y2  -j-  12  V  2  =  18 y  2 

Reducing  them  to  their  sim- 
plest form,  we  have   3V^8  =  6-^/2,  and  4y^i8  =  12-^/2,   which  are 
like  radicals.    (Arts.  302,  305.)    Now  6y'2  and  12-^/2  =  18-^/2,  Ans. 

309.  How  reduce  quantities  to  any  required  index?  Note.  To  what  is  this  oper- 
ation simiJar  ? 


160  SUBTRACTION     OF     RADICALS. 

3.  What  is  the  sum  of  3A/1S  and  4'V^24. 
Analysis.  —  Reducing  the  operation. 

radicals  to  tlieir  simplest  form,  3\/l8  =  '^'V/o  X  a/z  =  qV^ 

we  have   3  V^  =  9^2,   and  ^^-  ^  ^^g  x  'v/3  =  S^v^S 

4/^24  =  8/^3,  whicli  are  un-  ,—  3  ,- 

like  quantities,  and  can  only  ^^^-   9V  2  +  Ws 

be  added  by  writing  tliem  one  after  the  other,  with  their  proper  signs. 
(Arts.  43,  67.)    Hence,  the 

Rule. — I.  Reduce  the  radicals  to  their  simplest  form. 

II.  If  the  radical  parts  are  aliJcCf  add  the  coefficieiits,  and 
to  the  sum  annex  the  common  radical. 

If  the  radicals  are  unliTce,  write  them  one  after  another, 
with  their  proper  signs. 

Note. — To  determine  whether  radicals  are  alike,  it  is  generally 
necessary  to  reduce  them  to  their  simplest  form.    (Art.  305.) 

Find  the  sura  of  the  following  radicals : 

4.  '\/i2  and  \/27.  9.  3V^  and  4V^i28. 

5.  V20  and  V48.  10.  7^/243  and  5  A/363. 

6.  2\^¥  and  ^\^a^.  11.  «V8i^  and  3«'v/49^. 

7.  a^/2,a%  and  cA/276/A  12.  dV^S^.smd  Vs^^c, 

8.  3Vi8«a;2  and  2^/32^^.  j^,  41^/^  and  5^/^. 


SUBTRACTION    OF    RADICALS. 

311.  To  Find  the  Difference  between  two  Radicals. 

I.  From  3V45  subtract  2 A/20. 
Analysis. — Reducing  to  the  operation. 

simplest  form,  we  have  gy^s  3'\/45  =  3^9  X  V^  =  gVs 

and    4  a/ 5.    which    are    like                /—             r          r  r 

.^     ^^          r         r         2V20Z11  2V4  X  V5  =4V5 
quantities.    Now  9 y  5  —  4^/5  ■ 

=  5a/5,    the    difference    re-  .•.3^45-2^20  =  5^/5 

quired.     Hence,  the 

310.  How  ad4  radioals  ?    iVpfe.  How  determine  whether  tUey  are  like  quantities  I 


MULTIPLICATION     OF     RADICALS.  161 

EuLE. — Reduce  the  radicals  to  their  simplest  form  ;  change 
the  sign  of  the  subtrahend,  and  proceed  as  in  addition  of 
radicals,     (Art.  310.) 

(2.)  (30_  (4.)_ 

From       4  a/772  V480  4  A/320 

Take  V44^  4  A/63  —  5-A/80 


5.  From  3a/49«S  take  2A/25fit^. 

6.  From  ^^/a  +  ^  take  3V^«  +  5. 

7.  From  3a/^  take  —  4a/^. 

8.  From  3V^25o^%  take  2'V^54J%. 

9.  From  —  a~^  take  —  2^"^. 
10.  From  5a/J  take  2a/J. 


MULTIPLICATION    OF    RADICALS. 
312.  To  Multiply  Radical  Quantities. 

1.  What  is  the  product  of  3a/«  by  2  a/^. 
Analysis. — Since  these  radicals  are  operation. 

of  the  same  degree,  we  multiply  the  3a/^  X  2\^b  ■=  6\^ab 

radical   parts    together,  like    rational 

quantities,  and  to  the  result  prefix  the  product  of  the  coefficients. 

2.  Multiply  3a/^  by  2'\/c. 

AiiTALYSTS. — As  these  radicals  are  of  differ-  opbbation. 

ent  degrees,  they  cannot  he  multiplied  together  3  ^/a  =z  3  (ai) 

in  their  present  form.     We  therefore  reduce  3/—  /   2\ 

2  A/  C  '~~  2  (  C^  / 

them  to  a  common  index,  and  then,  multiply-  ^     ^ 


ing  as  before,  we  have  6^a^c\  ^^«-   ^  {a^cP)^ 

3.  Multiply  «3  by  a^. 

Analysis. — These  radicals  are  of  different  degrees,  but  of  the  same 
radical  part  or  root ;  we  therefore  multiply  them  by  adding  their 
fractional  exponents.    ^  + 1  =  f .    Therefore,  a^  x  a^  =  a^.    Hence,  the 

311.  How  subtract  radicals  f 


162  MULTIPLICATION     OF     RADICALS. 

Rule. — I.  Reduce  the  radicals  to  a  common  index, 
II. — Multiply  the  radical  parts  together  as  rational  quan- 
tities, and  placing  the  result  under  the  common  index,  prefix 
to  it  the  product  of  the  coefficients. 

Notes. — i.  Roots  of  like  quantities  are  multiplied  together  by 
adding  their  fractional  exponents.     (Art,  94.) 

2.  This  rule  is  based  upon  the  principle  that  the  product  of  the 
roots  of  two  or  more  quantities  is  the  same  as  the  root  of  the  product. 
(Art.  293,  Prin.  3.) 

3.  The  product  of  radicals  becomes  rational^  whenever  the  numer- 
ator of  the  index  can  be  divided  by  its  denominator  without  a 
remainder. 

4.  If  rational  quantities  are  connected  with  radicals  by  the  signs  + 
or  — ,  each  term  in  the  multiplicand  must  be  multiplied  by  each  term 
in  the  multiplier.    (Art.  98.) 

Multiply  the  folkwing  radicals: 

4.  5a/i8  by  3V20.  10.  OP-  by  q>, 

5.  a\/x  by  h^fx,  11.  7V^4  by  3^4. 

6.  ^/a  +  i  by  V«  —  ^.        12.  V9«  by  Vi6a. 

7.  ^fax  by  ^fcy,  13.  \/i8  by  ^/ 2. 

8.  c^  by  (^,  14.  "^Zax  by  ^fwx, 

(9-)  (X5-) 

Multiply    a  -f-  V?  Mult.    «  +  \/x 

By  g-f-  V5  By         i  +  l\/x 

ac  +  cVi  a  +  Va; 

+  aVd  +  v^  -t-  ahVx  -f  Ja: 


J/i5.  «c  +  cA/^-f  «A/^-f  V^       Ans.  a-{-V^-\-ahVx-^bx 

16.  2\/f  by  2\/J.  18.     (m-]-n)^hj{m-\'n)^ 

17.  4V|by3V|.  ^^^     J^^^J~^_. 


313.  How  multiply  radicals?  iVbfe*.  How  arc  roots  of  like  quantities  multi- 
plied? Upon  what  principle  is  this  rule  based?  When  does  the  product  of 
radicalB  become  r'vtional?  If  radicals  are  connected  with  rational  quantities,  how 
multiply  them? 


DIVISION"     OF     RADICALS.  163 

DIVISION    OF    RADICALS. 
313.  To  Divide  Radical  Quantities. 

1.  Diyide  4^/^400  by  2\/Sa. 

Analysis. — Since  the  given  radicals  are  operation. 

of  the  same  degree,  one  may  be  divided  by  4\/T4ac            / — 

the  other,  like  rational  quantities,  the  quo-  /^—    ^^  2V3C 

2  V  o<^ 

tient  being  Y  3c.     (Art.  iii.)     To  this  result 

prefixing  the  quotient  of  one  coeflBcient  divided  by  the  other,  we  have 
2  -v/sc,  the  quotient  required. 

2.  Divide  4 Vac  by  2  Vet. 

Analysis. — Since  these  radicals  are  aVoc       a  (ac)^ 

of    different    degrees,   they    cannot    be  — rp-  = ^ 

divided    in    their  present    form.      We  2\/a  2  [a)^ 

therefore  reduce    them    to    a  common  4\/ttC       4  (aW^ 

index,  then  divide  one  by  the  other,  and         •*•       s/~~  ^^        ,  „.  1 

2  A/  (l  2  \  (I  I  ^ 

to  the  result  prefix  the  quotient  of  the  ^  ^    ' 

coefficients.    Tlie  answer  is  2^0^.  or  2  (a(?)^,  Ans. 

.  3.  Divide  a^  by  a^ 

Analysis. — These  radicals  are  of  different  opbeation. 

degrees,  but  have  the  same  radical  part  or  root.  ^h  —  ^1 

We  therefore  divide  them  by  subtracting  the  1. 2 

fractional  exponent  of  the  divisor  from  that  of 

the  dividend.  (Art.  113.)  Reducing  the  expo-  a^  —  a^  z=:  a^ 
nents  to  a  common  denominator,  a^  =  a^,  and  .  ^^  .  /^^  — ■  rA 
as— a«.    Now  a^-i-a^=a^,  Ans.    Hence,  the 

Rule.— I.  Reduce  the  radical  parts  to  a  common  index. 

11.  Divide  one  radical  part  hy  the  other,  and  placing  the 
quotient  under  the  common  index,  prefix  to  the  result  the 
quotient  of  their  coefficients. 

Note. — Boots  of  like  quantities  arfe  divided  Tyy  subtracting  thefrat- 
tional  exponent  of  the  divisor  from  that  of  the  dividend.     (Art.  T13.) 

313.  How  divide  radicals  ?    2^ote.  How  divide  roots  of  like  quantities  ? 


164  IKVOLUTIOK     OF     KADICALS. 

Divide  the  following  radicals : 
4.     ^/\2a^c  by  ^/ ^c.  10.     14a V^  by  7a/^. 


9 


d^Mx^  by  2^/dx.  11.  («  +  5)t  by  (a  +  Z>)«. 

(a^  -f  fl^ic)^  by  a^,  12.  3^500;^  by  v^. 

12  (ay)T  by  («y)i  13.  ^/x^  —  f-  by  ^/x-\-y, 

2^1^/ ax  by  S-v/flf.  14.  16^/32  by  2^/4. 

iS^cs/^rc  by  2cV^.  15.  8^512  by  4^/2. 


INVOLUTION    OF    RADICALS. 
314.  To  Involve  a  Radical  to  any  required  Power. 

I.  Find  the  square  of  a^, 

OPERATION. 

Analysis. — As  a  square  is  the  product  of  two  1  i  2 

Ct^    X  d^  "^^^  (X^ 
equal  factors,  we  multiply  the  given  index  by 

the  index  of  the  required  power.    Hence,  the  .*.  0^,   Ans. 

EuLE. — Multiply  the  index  of  tlie  root  'by  the  index  of  tlie 
required  power,  and  to  the  result  prefix  the  required  poivei 
of  the  coefficient. 

Note. — A  root  is  raised  to  a  power  of  the  same  name  by  removing 
the  radical  sign  or  fractional  exponent.    (Ex.  2.) 

2.  Find  the  cube  of  ^/a  +  J.  Ans.  a  ■\-b. 

3.  Find  the  cube  of  a^, 

4.  What  is  the  square  of  3a/2^. 

5.  What  is  the  cube  of  2\^. 

X     ■ 

6.  Eequired  the  cube  of  -V2X, 

7.  Required  the  cube  of  4\  /  — • 

V    4 

8.  Find  the  fourth  power  of  3A  /  — 

9.  What  is  the  square  of  «  +  \/y  ? 

314.  How  Involve  radicals  to  any  required  power  ?  Note.  How  raise  a  root  to  a 
yower  of  the  same  name? 


BVOLUTIOK     OF     RADICALS.  165 

EVOLUTION    OF    RADICALS. 
315.  To  Extract  the  Boot  of  a  RadicaL 

I.  Find  the  cube  root  of  a^'i/W. 

Analysis. — Finding  the  root  of  a  radi-  operatiok.    

cal  is  the  same  in  principle  as  finding  the  V^a^V^^  =   v  ^3^1 

root  of  a  rational   quantity.      (Art.  296.)  3 

Reducing  the  index  of  the  radical  to  an  V^^^t  =:  ab^,  Ans. 

equivalent  fractional  exponent,  we  extract 

the  cube  root  by  dividing  it  by  3.    The  result  is  a&^.    Hence,  the 

Rule. — Divide  the  fractional  exponent  of  the  radical  by 
the  number  denoting  the  required  root,  and  to  the  result 
prefix  the  root  of  the  coefficient. 

Notes. — i.  Multiplying  the  index  of  a  radical  by  any  number  is  the 
same  as  dividing  the  fractional  exponent  by  that  number. 

Thus,  /^a  =  aK  Multiplying  the  former  by  2,  and  dividing  the 
latter  by  2,  we  have  ^a  =  a". 

2.  If  the  coefficient  is  not  a  perfect  power,  it  should  be  placed  under 
the  radical  sign  and  be  reduced  to  its  simplest  form.     (Art.  305.) 

2.  Required  the  square  root  of  gV^, 

3.  Required  the  square  root  of  4\^^. 

4.  Find  the  cube  root  of  3a/^. 

5.  Find  the  cube  root  of  2bV2b. 

6.  What  is  the  cube  root  of  a  (bc)^  ? 

7.  What  is  the  fourth  root  of  ^^f  ? 

8.  What  is  the  fourth  root  of  Va^  V^^  ? 

9.  Find  the  seventh  root  of  i28a/«. 
10.  Find  the  fourth  root  of  Vayby, 

II.  Find  the  fifth  root  of  4a^V2a. 
12.  Find  the  nth  root  of  aVbc. 

315.  How  extract  the  root  of  a  radical  ?  Notes.  To  what  is  multiplyinff  the  index 
Of  a  radical  equivalent  ?    If  the  coefficient  is  not  a  perfect  power,  what  is  dope  ? 


166     CHAI^GIKG    RADICALS    TO    llATIOKALS. 


CHANGING    A 


RADICAL    TO 
QUANTITY. 


A    RATIONAL 


CASE    I. 
316.  To  Change  a  Radical  Monomial  to  a  Rational  Quantity 

1.  Change  V^  ^^  ^  rational  quantity. 

Analysis. — Since  multiplying  a  root  of  a 
quantity  into  itself  produces  the  quantity,  it 
follows  that  /y/a  x  ^ a  =  a,  which  is  a  rational 
quantity.     (Art.  287.) 

2.  It  is  required  to  rationalize  c^. 


OPERATION. 


OPERATION. 


Analysis. — A  root  is  multiplied  by  another 
root  of  the  same  quantity  by  adding  the  expo-  ^i  y^  ^1  -^^  ^ 

nents  ;  therefore  we  add  to  the  index  \  such  a 
fraction  as  will  make  it  equal  to  i.    (Art.  94.) 

Thus,  as  X  as  =  as+ s  =  fl^i  =  «^  the  rational  quantity  required. 

3.  It  is  required  to  rationalize  ^. 

Solution.— Multiplying  a^  by  ic«,  the  result  is 
a?,  which  is  a  rational  quantity.    Hence,  the 


OPERATION. 


Rule. — Multiply  the  radical  hy  the  same  quantity  having 
such  a  fractional  exponent  as,  when  added  to  the  given 
expo7ient,  the  sum  shall  be  equal  to  a  unit,  or  i. 


4.  Required  a  factor  which  will  rationalize  a^. 

5.  What  factor  will  rational 

6.  What  factor  will  rational 

7.  What  factor  will  rational 

8.  What  factor  will  rational 

9.  What  factor  will  rational 
xo.  What  factor  will  rational 


ze  Va^c? 
ze  '^(a  +  by. 
ze  Va^? 
ze  ^^+jY? 
ze  \/(a+  by? 
ze  V{a  +  b  +  c)? 


316.  How  reduce  ft  rftdicftl  monomifll  to  »  r^Uonal  quantity? 


CHANGING     RADICALS     TO     RATIONALS.      167 


CASE     II. 
317.  To  Change  a  Radical  Binomial  to  a  Rational  Quantity. 

1.  Ifc  is  required  to  rationalize  ^s/a  +  ^/}). 
Analysis, — The  product  of  the  sum  and  opbratiok. 

difference  of  two  quantities  is  equal  to  the  /y/^  _i-  ^J}) 

difference  of  their  squares  (Art.  103);  there-  r-           ,j- 

fore  (/;/«+ \/&)  multiplied  by  {'s/a—^h) 

=  a—b,  which  is  a  rational  quantity.  a  +  ^/ob 

Therefore,  the  factor  to  employ  as  a  multi-  —  yai  —  b 

plier  is  V«-  V^-  a  —  b,  AnsT 

2.  What  factor  will  rationalize  Vx  —  Vy  ? 

Analysis.— If  the  binomial  ^x  —  y'y  is  operation.^ 

multiplied  by  the  same  terms  with  the  sign  of  Vcc  —  Wy 

the  latter  changed  to  + ,  we  have  w'~    .    ^T. 

(\/x  -  Y^y)  X  {^^x+  ^y)  =  x-y.  ~~~       ~ 

_           _  JO  *""•  y 

(Art.  103.)     Therefore,  y\/x  +  ^/y  is  the  fac-  ,-            .- 

tor  required.     Hence,  the  ^^^-    V^+Vy 

Rule. — Multiply  the  binomial  radical  by  the  correspond- 
ing binomial  with  its  connecting  sign  changed, 

3.  What  factor  will  rationalize  x  -\-  4  V9  ? 

4.  Rationalize  A/9  —  V^. 

5.  What  factor  will  rationalize  VT  +  Vet  ? 

6.  Rationalize  6  —  V^. 

7.  What  factor  will  rationalize  Vsa  —  V^b  ? 

8.  Rationalize  Vci'  —  Vs. 

9.  What  factor  will  rationalize  3  's/a  +  a/S  ? 
10.  What  factor  will  rationalize  4  V^a  —  5  V^? 

317.  How  reduce  a  radical  binomial  to  a  rational  quantity  ? 


168  RADICAL     FRACTIONS. 


CASE     III. 


318.  To  Change  a  Radical  Fraction  to  one  whose  Numerator 
or  Denominator  is  a  Rational  Quantity. 

1.  Change-^ to  a  rational  denominator. 

Analysis. — Multiply  both  terms  of  the  operation. 

fraction  by  the  denominator  A^h,  and  the  a     y^  ^^        aVb 

result    is    —x— ,     whose    denominator  is  V ^  X  yb  ^ 

rational.     (Art.  167,  Prin.  3,  note.)    Hence,  the 

Rule. — Multiply  both  terms  of  the  fraction  ly  such  a 
factor  as  will  malce  the  required  term  ratio7ial. 

Note — Since  the  product  of  the  sum  and  difference  of  two  quan- 
tities is  equal  to  the  difference  of  their  squares,  when  the  radical 
fraction  is  of  the  form  — =,  if  we  multiply  the  terms  by 

(y'a  +  -y/ft),  we  have  a  —  &  for  the  denominator.    (Art.  103.) 

2.  Rationalize  the  denominator  of  -~z»       Ans.  - — ^. 

Va   ^  ^  a 


3.  Rationalize  the  numerator  of  ---•  Ans, 


^/x  Vox 


4.  Rationalize  the  denominator  of  -■ 

x 


5.  Rationalize  the  denominator  of  — -• 

6.  Rationalize  the  denominator  of  —^ —• 

^x  —  Vy 

Of 

7.  Rationalize  the  denominator  of  —j= —* 

I 

I  +  V3 

3  — V3 


8.  Rationalize  the  denominator  of 

9.  Rationalize  the  denominator  of 


318.  How  reduce  a  radical  fraction  to  one  whose  numerator  or  denominator  is  a 
rational  quantity?  \^'^len  the  fractions  contain  compound  quantities,  what  prin- 
ciple  enters  into  their  reduction? 


BADICAL    EQUATIOI^S.  169 


RADICAL    EQUATIONS. 

319.  A  Madical  Bquation  is  one  in  whiCh  the 
unknown  quantity  is  under  the  radical  sign. 

320.  To  Solve  a  Radical  Equation. 
I.  Given  a/^  4-2  =  7,  to  find  x. 

Analysis. —  Transposing    2,    we  have,  y5  +  2  =  7 

/\/x  =  5.    Since   5   is  equal  to  the  >\/x,  it  /— 

follows  that  the  square  of  5,  or  25,  must  be  V     —  7  —     —  5 
the  square  of  y\/x.    Therefore,  x  =  25. 


a;  =  52  =  25 


2.  Given  2a  -{-  \/x  =  ()a,  to  find  x. 

Solution. — ^By  the  problem,        la  +  ^Jx  —  t^n 

By  transposing,  ^x  =  ya 

By  involution,  x  =  49a' 

3.  Given  5V^a;  +  i  =  35,  to  find  x. 

Solution.— By  the  problem,         5/y/a;  +  i  =  35 

Removing  coefficient,  ^^x  +  i  =     7 
Involving,  a?  +  i  =  343 

Transposing,  X  =  342.     Hence,  the 

Rule. — Involve  both  sides  to  a  power  of  the  same  name  as 
the  root  denoted  by  the  radical  sign. 

Note. — Before  invohing  the  quantities,  it  is  generally  best  to  clear 
of  fractions,  and  transpose  the  terms,  so  that  the  quantities  under  tJ.a 
radical  sign  shall  stand  alone  on  one  side  of  the  equation. 

Eeduce  the  following  radical  equations: 


4.     «  H-  v^  -\-  c  =  d. 

8.     v^22;  +  3  —  6  =  13. 

5.       ^/X+  2z=S. 

9.     v"^  —  4  =  3- 

6.    sVa;  — 4  4-S  =  7i• 

10.     2^a;  — 5=4. 

7•     3a/^=24. 

II.     5A^  =  30. 

319.  ^  hat  is  a  radical  equation  ?   320.  flow  solved  ?    Note.  What  should  be  done 
before  involving  the  quantities  ? 

8 


170  RADICAL     EQUATIONS, 


12.  Given     ^  "^^ — -i—  =  J2,  to  find  iR 

13.  Reduce  a/cl^  +Vx=z    ^/"^^^    . 

Analysis. — By  removing  tlie  opkkation. 

denominator  the  first  member  is  /~Z          7=^               3  +  ^ 

squared.    But  x  is  still  under  y         -r  V  ^           .  /                  v 
the    radical  sign.    This  is  re- 

moved  by  involving  both  mem-  ^  +  v  ^  =  3  +  ^ 

bers  again.  Atis.  X  =  ($ -\-  C  —  fl') 

14.  Given  ^^^  =  :^,  to  find  ST. 


2^2 


15.  Given  x  +  a/«^  4-  ic^  =  _______     to  find  a;. 

Note. — If  the  equation  has  two  radical  expressions,  connected  with 
other  terms  by  the  signs  +  or  — ,  it  is  advisable  to  transpose  the  terms 
so  that  one  of  the  radicals  shall  stand  alone  on  one  side  of  the  equation. 
By  involving  both  members,  one  of  the  radicals  becomes  rational ;  and 
by  repeating  the  operation,  the  other  will  also  disappear. 


16.  Given  V4  +  5.'^^  —  ^z^  =--  2,  to  find  x. 
Transposing,  'Y/4  +  S-c  =  ^/3x  +  2 

Involving,  4  +  5«  =  4  +  4\/3'«  +  3« 

/ —       ^ 
Transiiosing  and  dividing  by  4,  ysx  —  - 

Involving,  ^x  =  — 

4 

Transposing  and  multiplying  by  4,         x'^  =  I2j; 
Hence,  x  =  12,  Ana. 

17.  Given  a/^  -f  12  r=:  2  +  v^,  to  find  x. 

18.  Given  Vs  x  Vx  4-2  =  2  +  Vs^,  to  find  Xo 

^.         Vx       X  —  ax    ,     „    , 
10.  Given = — — ,  to  find  x, 

(See  Appendix,  p.  289.) 


CHAPTER    XV. 
QUADRATIC    EQUATIONS. 

321.  Equations  are  divided  into  different  degrees,  as  the 
fii-st,  second,  third,  etc.,  according  to  the  powers  of  the 
unknown  quantity  contained  in  them. 

An  equation  of  the  First  Degree  is  called  a  Simple 
Equation^  and  contains  only  the /r*^  power  of  the  unknown 
quantity. 

An  equation  of  the  Second  Degree  is  called  a  Quad- 
vatic  Equation,  and  the  highest  power  of  the  unknown 
quantity  it  contains  is  a  square. 

An  equation  of  the  Third  Degree  is  called  a  Cuhic 
EquoMon,  and  the  highest  power  of  the  unknown  quantity 
it  contains  is  a  cul:)e. 

An  equation  of  the  Fourth  Degree  is  called  a 
Biquadratic,  etc. 

322.  Quadratic  Equations  ai-e  divided  into  pure 

and  affected. 

323.  A  Pure  Quadratic  contains  the  square  only  of 
the  unknown  quantity ;  as,  x^  =  5. 

324.  An  Affected  Quadratic  contains  both  the  first 
and  second  powers  of  t h  e  unkn  own  quantity ;  as,  x^-\-ax= cd. 

Notes. — i.  Pure  quadratics  are  sometimes  called  incomplete  equa- 
tions ;  and  affected  quadratics,  complete  equations. 

321.  How  are  equations  divided  ?  What  is  an  equation  of  the  first  de^^ree  ?  The 
second?  Third?  Fourth?  322.  How  are  quadratic  equations  divided  ?  325.  What 
I?  h  pure  quadratic  ?  324.  An  afl;"ected  quadratic  ?  Note.  What  are  they  sometimes 
called  ? 


172  PURE     QUADRATICS. 

2.  A  Complete  Equation  contains  every  integral  power  of  tlie  un- 
known quantity  from  that  which  denotes  its  degree  down  to  the  zero 
power. 

An  Incomplete  Equation  is  one  which  lacks  one  or  more  of  these 
powers. 


PURE    QUADRATICS. 

325.  Every  pure  quadratic  may  be  reduced  to  the  form 

a;3  =  «. 

For,  by  transposition,  etc.,  all  the  terms  containing  x^  can  be 
reduced  to  one  term,  as  lyx^ ;  and  all  the  known  quantities  to  one 
term,  as  c.    Then  will 

W  =  c. 

Dividing  both  members  by  &,  and  substituting  a  for  the  quotient  of 
e-r-b,  the  result  is  the  form, 

x^  =  a. 

326.  Pure  quadratic  equations  have  two  roots,  which  are 
the  same  numerically,  but  have  opposite  signs.  (Art.  293, 71.) 

Thus,  the  square  of  +a  and  of  —a  is  equally  a^.    Hence, 

V^=  ±a, 

327.  To  Solve  a  Pure  Quadratic  Equation. 

I,  Find  the  value  of  a;  in 6  = h  2, 

9  3 

Solution.— Given 6  =  —  +  2 

9  3 

Clearing  of  fractions,         5a^  —  54  =  3^^  +  l8 
Transposing,  etc.,  2x'^  =  72 

Removing  coefficient,  a^  =  36 

Extracting  sq.  root,  x  =  ±6,  Ans, 

Substituting  b  for  2,  and  c  for  72,  in  the  third  equation,  we  have 
the  form,  bx^  =  c. 

Removing  coefficient,  etc.,      x^  =  a.      Hence,  the 

Rule. — Reduce  tlie  given  equation  to  the  form  x^  =  a,  and 
extract  the  square  root  of  holh  members.     (Art.  296.) 

336.  Hqw  manjT  roots  nas  a  pure  ooa^raticf 


PURE     QUADRATICS.  173 

Find  the  value  of  x  in  the  following  equations : 

2.  3^5^ —  5  =  70.  10.  20^  +  12  ^  3a;2_  27. 

3.  92?^  +  8  =  3a;2 -f- 62.  11.  72^2  —  7  =  32^2  +  9. 

4.  5a;2  -I-  9  =  2a;2  ^  27.  12.  ahy^  =  a*. 

5.  6a«  +  5  =  40?^  +  55.  13.  {x  +  2)2  =  4a;  +  5. 

6.  ^  +  35  =  3^+7.  14.     ii^-i=^^^.     - 
4  4 

£^±8_^-6  ^(2^+9)  __  3^-1-6 

8.  -  = ".  16.     —5 1 f-  =  -. 

42a;  4  —  a;4H-a;3 

:r  .   2      a;  .    ^  ax^ia  —  2) 

9.  -  +  -  =  -  +  -•  17.    — ~T =  I  —  ic. 

328.  Radical  equations,  when  cleared  of  radicals,  often 
become  pure  quadratics. 

18.  Given  Vx^  +  n  =  \/2x^  —  5,  to  find  a:. 

Solution.— Clearing  of  radicals,  aj2  +  ii  =  2aJ'  —  5 

Transposing  and  extracting  root,  jc  =  ±  4 


4a; 


19.  Given  2V^  — 5  =  — ,  to  find  ar. 


20.  Given  2V^--4  =  4V^---i,  to  find  a?. 

21.  Given  ^/^"-M  =  ,  to  find  a?. 

V  a;  —  ^ 

22.  Given  \/- — ^^^  =  Vx,  to  find  a; 

23.  Given  =  'v/iTT^,  to  find  x. 

24.  Given  ■  =  \/x  —  lo,  to  find  x, 

vx  +  10 

327,  What  is  the  rule  for  the  solution  of  pure  quadratics?    328.  What  may 
radical  equations  become  ? 


174  PUEB    QUADKATIOS. 


PROBLEMS 


1.  The  product  of  one-third  of  a  number  multiplied  by 
one-fourth  of  it  is  io8.     What  is  the  number  ? 

2.  What  number  is  that,  the  fourth  part  of  whose  square 
being  subtracted  from  25,  leaves  9  ? 

3.  How  many  rods  on  one  side  of  a  square  field  whose 
area  is  10  acres  ? 

4.  A  gentleman  exchanges  a  rectangular  piece  of  land 
50  rods  long  and  18  wide,  for  one  of  equal  area  in  a  square 
form.    Kequired  the  length  of  one  side  of  the  square. 

5.  Find  two  numbers  that  are  to  each  other  as  2  to  5,  and 
whose  product  is  360. 

6.  If  the  number  of  dollars  which  a  man  has  be  squared 
and  7  be  subtracted,  the  remainder  is  29.  How  much 
money  has  he  ? 

7.  Find  a  number  whose  eighth  part  multiplied  by  its 
fifth  part  and  the  product  divided  by  16,  will  give  a  quotient 
of  10. 

8.  The  product  of  two  numbers  is  900,  and  the  quotient 
of  the  greater  divided  by  the  less  is  4.  What  are  tlie 
numbers  ? 

9.  A  merchant  buys  a  piece  of  silk  for  I40.50,  and  the 
price  per  yard  is  to  the  number  of  yards  as  3  to  54. 
Required  the  number  of  yards  and  the  price  of  each. 

10.  Find  a  number  such  that  if  3  times  the  square  be 
divided  by  4  and  the  quotient  be  diminished  by  12,  the 
remainder  will  be  180. 

11.  A  reservoir  whose  sides  are  vertical  holds  266,112 
gallons  of  water,  is  6  feet  deep,  and  square  on  the  bottom. 
Required  the  length  of  one  side,  allowing  231  cubic  inches 
to  the  gallon. 

12.  What  number  is  that,  to  which  if  10  be  added,  and 
from  which  if  10  be  subtracted,  the  product  of  the  sum  and 
difference  will  be  156  ? 


AFFECIEB     QUADBAIICS  175 


AFFECTED    QUADRATICS* 

329.  An  Affected  Quadratic  Equation  is  one 

which  contains  the^rs^  and  second  powers  of  the  unknown 
quantity ;  as,  aoi?  -^  Ix  z=i  c, 

330.  Every  affected  quadratic  may  be  reduced  to  the  form, 

in  which  a,  b,  and  x  may  denote  any  quantity,  either 
positive  or  negative,  integral  ov  fractional. 

For,  by  transposition,  etc.,  all  the  terms  containing  x^  can  be 
reduced  to  one  term,  as  cx"^ ;  also,  those  containing  x  can  be  reduced 
to  one  term,  as  dx ;  and  all  containing  the  known  quantities  can  be 
reduced  to  one  term,  as  g.    Then,  cx^  +  dx  =  g. 

Dividing  both  members  by  c,  and  substituting  a  for  the  quotient  of 
d-r-c,  and  b  for  the  quotient  of  g  -i-  c,  we  have, 
a^  +  ax  =  b. 

Take  any  numerical  quadratic,  as-^ S  =  x^  +  —  —  4, 

Clearing  of  fractions,  Saj^  —  4a;  —  48  =  6aJ*  +  4a;  —  24 

Transposing,  etc.,  2af^  —  Sx  =  24 

Removing  the  coefficient,  a^  —  4X  =  12 

Substituting  a  for  4,  and  6  for  12  in  the  last  equation,  we  have, 
x^  —  ax  =  b.    Hence, 

All  affected  quadratics  mag  be  reduced  to  the  general  form, 
%^  ±^ax  =  b, 

331.  The  First  Member  of  the  general  form  of  an 
affected  quadratic  equation,  it  will  be  seen,  is  a  Bi?iomial, 
but  not  a  Complete  Square.  One  term  is  tvanting  to  make 
the  square  complete.  (Art.  266,  note.)  The  equation, 
therefore,  cannot  be  solved  in  its  present  state. 


329.  What  is  an  affected  quadratic  equation  ?  330.  To  what  general  form  may 
every  affected  quadratic  be  reduced?  331.  What  is  true  of  the  first  member  of  the 
general  form  of  an  affected  quadratic  ? 

*  Quadratic, ivGm  the  Latin  qtiadrare^  to  make  square. 
Affected,  made  up  of  different  powers  ;  from  the  Latin  ad  and/acw>, 
to  make  or  join  ta 


176  AFFECTED     Q  tJ  ADHATICS. 

332.  There  are  three  methods  of  completing  the  square 
and  solving  the  equation. 


FIRST    METHOD. 

I,  Given  a:'  +  2aa;  =  J,  to  find  the  value  of  x. 

Analysis. — The  first  opeeation. 

and  third  terms  of  the  01^  -{-  2ax  =.  b 

Bquare  of  a  binomial  are  a?  -j-  2ax  •\-  d^  =.  a^  -{■  h 
complete  powers,  and  the  x  4-  a  =  -h  a/^2   i    a 

second  term  is  twice  the 


product  of   their  roots; 

or  the  product  of  one  of  the  roots  into  twice  the  other.    (Art.  loi.) 

In  the  expression,  x'^  +  2ax,  the  first  term  is  a  perfect  square,  and 
the  second  term  2ax  consists  of  the  factors  2a  and  x.  But  x  is  the 
root  of  the  first  term  x'^ ;  therefore,  the  other  factor  2a  must  be  tmce 
the  root  of  the  third  term  which  is  required  to  complete  the  square. 
Hence,  half  of  2a,  or  a,  must  be  the  root  of  the  third  term,  and  a^  the 
term  itself.  Therefore,  x-  +  2ax  +  a^  is  the  square  of  the  first  member 
completed. 

But  since  we  have  added  a^  to  the  first  member  of  the  equation, 
we  must  also  add  it  to  the  second,  to  preserve  the  equalitj. 
Extracting  the  square  root  of  both  members,  and  transposing  a,  we 
have  x  =  ~a±  ^/a'^  +  b,  the  value  sought.   (Art.  297.) 

2.  What  is  the  value  of  a;  in  20^  +  x  =  64  —  yx? 

Analysis.  —  'transposing     —  70?  operation. 

and  removing  the  coefficient  of  a^,  2X  +  x  =z  64.  —  'jx 

we  have  the  form  x^  +  4X  =  32.    But  23^  -}-  8x  =  64 

the  first  member,  x'^  +  4X,  is  an  incom-  a^  -{-  4X  =z  22 

plete  square  of  a  binomial.  x^  4-  ax  4-  A  ^=  '16 

In  order  to  complete  the  square, 
we  add  to  it  the  square  of  half  the 
coefficient  of   x.    (Art.  266.)    Now, 

having  added  4  to  one  member  of  *•  €.,  X  =  4  OT  —  8 
the  equation,  we  must  also  add  4  to 

the  other,  to  preserve  their  equality.  Extracting  the  root  of  both, 

and  transposing,  we  have  a;  =  4,  or  —  8.  (Art.  297.) 

333.  How  many  methods  of  completing  the  square  ? 


a?+  2  =  ±  6 

X=  —  2  ±6 


AFFECTED     QUADRATICS.  177 

Notes.— I.  Adding  the  square  of  half  the  coefficient  of  the  second 
term  to  both  membei-s  of  the  equation  is  called  completing  the  square. 

2.  The  first  member  of  the  fourth  equation  is  the  square  of  a  bino- 
mial ;  therefore,  its  root  is  found  by  taking  the  roots  of  the  first  and 
third  terms,  which  are  perfect  powers.  (Art.  297.)  From  the  process 
of  squaring  a  binomial,  it  is  obvious  that  the  middle  term  (4a;)  forms 
no  part  of  the  root.    (Art.  266.) 

333.  From  these  illustrations  we  derive  the  following 

EuLE. — I.  Reduce  the  equation  to  the  form,  x^  ±ax  =  b. 

II.  Add  to  each  member  the  square  of  half  the  coefficient  ofx 

III.  Extract  the  square  root  of  each,  and  reduce  the  result 
ing  equation. 

3.  Find  the  value  of  a;  in  —  2:??  -f  2>ax  =  —  65. 

Solution. — By  the  problem,  —2a;'  +  ^ax  =  —6b 

Removing  coefficient  of  x^,  —x'^  +  ^ax  =  — 3& 

Making  x^  positive  (Art,  140,  Prin.  3),   x^—4ax  =  3& 
Completing  square,  a^—^ax + ^a^  —  4a'  +  3& 


Extracting  the  root,  x—2a  =  ±  Y4a'^  +  3ft 

.*.    x  =  2a±  ^\a?  +  38 
4.  Given  ^  ■\-  ax  -\- bx  zzz  d,  to  find  x, 

OPEBATION. 

ix?  ■{■  ax  -\- bx  =z  d 
a^+  (a  -\-b)x  =  d 

Analysis. — Factoring  the  terms  which  contain  the  first  power  of  x, 
we  have  ax  +  hx  =  {a  +  b)x  ;  hence,  (a  +  h)  maybe  considered  a  com* 
pound  coefficient  of  x.  By  adding  the  square  of  half  this  coefficient  to 
both  members,  and  extracting  the  root,  the  value  of  x  is  found. 

333.  What  is  the  rule  for  the  first  method  of  solving  affecj^d  quadratics » 


178  AFFECTED     QUADRATICS, 

5.  Given  32;  —  2ic2  _  _  ^^  ^0  ^^^  ^.^ 
Solution. — By  the  problem,  3X  —  2X^  =  —  g 


Making  x^  positive,  etc., 

x^- 

3X_ 
2 

.9 
2 

Completing  square. 

X'- 

f- 

9 
16 

2         16 

81 
~  16 

Extracting  root, 

X- 

_3  ^ 
4 

±9 
4 

/.  X  = 

f±f. 

i.  e. 

,  X  = 

:  3  or  — 

li 

6.  Given  $x^  —  24X  =  —  36,  to  find  x. 

Ans.    +  6  or  +2. 

Note, — The  two  Toots  of  an  affected  quadratic  may  have  the  same 
or  different  signs.  Thus,  in  the  6th  and  12th  examples  they  are  the 
same;  in  the  ist,  2d,  3d,  4th,  and  5th,  they  are  different. 

7.  Given  ^x^  —  40a;  =  45.  to  find  x. 

8.  Given  x^  —  6ax  =  d,  to  find  x. 

9.  Given  2X^  +  2ax  =  2  (J  -f  c),  to  find  x. 

Solution. — Completing  the  square,  x"^  +  ax  -^ —  = b  b  +  c, 

4        4 


Extracting  root,         x+-  =  ±-i/—  +  b  +  c 
2  'A 


Transposing,  3?= ±  y  — \-  b  +  c 

10.  Given  2X^  —  22a;  =  120,  to  find  x. 

11.  Given  x^  —  140  z=  13a;,  to  find  x. 

12.  Find  the  value  of  a;  in'  x^  —  ^x  ■{•  1  =  k^x  —  15. 

Solution. — By  the  problem,       a;^— 3a;  +  i  =  5a;— 15 
Transposing,  aj*— 8a;  =  — 16 

Completing  square,  a;'— 8a5+i6  =  o 

Extracting  root,  «— 4  =  o 

.*.    a;  =  4 

Note. — In  this  equation,  both  the  signs  and  the  numerical  values  of 
the  two  roots  are  alike.     Such  equations  are  said  to  have  equal  roots. 

i\ro<«,— What  signs  have  the  roots  of  an  affected  quadratic  f 


AFFECTED     QUADRATICS.  179 


SECOND    METHOD. 


334.   When  an  affected  quadratic  equation    has   been 
reduced  to  the  general  form, 

x^  -\-  ax  =  h, 
its  root  may  be  obtained  without  recourse  to  completing  the 
square. 

I.  Given  x^  -{-Zx=.  65,  to  find  x. 

Analysis. — After  the  square  operation. 

of  an  affected  quadratic  is  com-         /y3  a_  Q/y (\c 


pleted  and  the  root  extracted, 
the  root  of  the  third  term  is 


a;  =  —  4  ±  a/65T^6 
transposed  to  the  second  mem-  •*•  2;  =        4  i  9 

her,  by  changing  its  sign.    (Art.  i,e.y  X  =  ^  or  — 1 3 

204.) 

Now,  if  we  prefix  half  the  coeflScient  of  x,  with  its  sign  changed,  to 
plus  or  minus  the  square  root  of  the  second  member  increased  by  the 
square  of  half  the  coefficient  of  x,  the  second  member  of  the  equation 
will  contain  the  saiTie  combinations  of  the  same  terms,  as  when  the 
square  is  completed  in  the  ordinary  way.    Hence,  the 

Rule. — Prefix  half  the  coefficient  of  x,  with  the  opposite 
sign,  to  plus  or  miuus  the  square  root  of  the  second  member, 
increased  hy  the  square  of  half  the  coefficietit  of  x. 

Solve  the  following  equations : 


2. 

3^^  —  9^  —  3  =  207- 

8. 

a«  -f-  4ax  —  b. 

3- 

42^2+  120;  +  5=45. 

9- 

32)2  — 74  =  6a; +  31. 

4. 

ix^—  14a: -f-  15=0. 

10. 

a^  -\-  i^  =  6x. 

5. 

4a;2  _  pa;  __  28. 

II. 

{X  —  2)  {X—  l)=  20c 

6. 

2X             X  +  2 

12. 

x+  1           X     _is 
X       ^   x-\-\        6  ' 

7. 

a^  +  ^--ah  =  d, 
I? 

13. 

^^^-j^ch  =  bd. 
c 

334.  What  is  the  second  method  of  solving  affected  quadratics  ¥ 


180  AFFECTED     QUADRATICS. 


THIRD    METHOD. 

335.  A  third  method  of  reducing  an  affected  quadratic 
equation  may  be  illustrated  in  the  following  manner: 

1.  Given  a'3^  -^-Ixzzl  c,  to  find  x. 
Analysis.  —  Multiply-  operation. 

ing  the  given  equation  by  CL^  -\-  hx  ^z  c 

a,  the  coefBcient  of  a;^  and  /^a^a^  +  /[ahx  =  ^ac 

by  4,  the  smallest  square      ^^Z^s  _j_  ^al)X-\-  W  =  4«c  -f  J3 

number,  we  have  ^^«.   i    z>  i    .  / i — m 

4«2a;2  +  ^ahx  =  ^ac,  _L 

the  first  term  of  which  is  .    /„  _  — ^±V4^g  +  ^ 

an    exact  square,  whose  2a 

root  is  lax.       Factoring 
the  second  term,  we  have  /s^abx  =  2  (2aajx5).    (Art.  119.) 

As  the  factor  2.ax  is  the  square  root  of  4a^a;-,  it  is  evident  that  ^a^^ 
may  be  regarded  as  the  first  term,  and  /^cibx  the  middle  term  of  the 
square  of  a  binomial.  Since  ^ahx  is  twice  the  product  of  this  root  lax 
into  &,  it  follows  that  &  is  the  second  term  of  the  binomial  ;  conse- 
quently,  6'  added  to  both  members  will  make  the  first  a  complete 
square,  and  preserve  the  equality.  (Axiom  2.)  Extracting  the  square 
root,  transposing,  etc.,  we  have, 

X  = — — ,  the  value  of  x  required. 

2.  Given  2:1^  4-  3a;  =  27,  to  find  x. 

Solution. — By  the  problem,  2X^  +  3a;  =  27 

Multiplying  by  4  times  coef.  of  a;*,  i6ar'^  +  24a;  =  216 
Adding  square  of  3,  coef.  of  a?,  i6«2  +  24a; + 9  =  225 
Extracting  root,  4a;+3  =  ±  15 

Transposing,  425  =  —3  ±  15 

.\  flj  =  3  or  — 4|. 

336.  From  the  preceding  illustrations,  we  derive  the 
KuLE. — I.  Reduce  the  equalion  to  the  form,  aa^  ±_bx  =  c. 

II.  Multiply  both  members  hy  4  tiines  the  coefficient  of  a^. 

III.  Add  the  square  of  the  coefficient  of  x  to  each  member, 
extract  the  root,  and  reduce  the  resulting  equation. 

336.  What  is  the  rale  for  the  third  method  of  reduciug  affected  qoadrhtics  f 


AFFECTED     QUADRATICS.  181 

Notes. — i.  Wlien  the  coeflBcient  of  x  is  an  even  numbei  it  is 
sufficient  to  multiply  both  members  by  the  coefl&cient  of  x^,  and  add 
to  each  the  square  of  half  tlie  coeflScient  of  x. 

2.  The  object  of  multiplying  the  equation  by  the  coefficient  of  x^  is 
to  make  the  first  term  a  perfect  square  without  removing  the  coefficient. 
(Art.  251.) 

3.  The  reason  for  multiplying  by  4,  is  that  it  avoids  fractions  in 
completing  the  square,  when  the  coefficient  of  a;  is  an  odd  number. 
For,  multiplying  both  members  by  4,  and  adding  the  square  of  the 
entire  coefficient  of  x  to  each,  is  the  same  in  effect  as  adding  the  square 
of  half  the  coefficient  of  x  to  each,  and  then  clearing  the  equation  of 
fractions  by  multiplying  it  by  the  denominator  4. 

4.  This  method  of  completing  the  square  is  ascribed  to  the  Hindo<^ 

3.  Given  ^a^  -\-  4X  =  39,  to  find  x,      Ans.  3  or  —  4^. 

Reduce  the  following  equations : 

4.  x^  —  $0=:  —X.  8.  2aj2  —  6a;  =  8. 

5.  5^ -f  32)2  _  2.  9.  32^2  4- 5a;  =  42. 

6.  4X^  —  'jx  —  2  =  o.  10.  a^  —  15a;  =  —  54. 

7.  50^2^  2a;  =  88.  II,  92^— 7a;  =116. 

337.  The  preceding  methods  are  equally  applicable  to  all 
classes  of  affected  quadratics,  but  each  has  its  advantages  in 
particular  problems. 

The  first  is  perhaps  the  most  natural,  being  derived  from 
the  square  of  a  binomial ;  but  it  necessarily  involves  frac- 
tio7iSy  when  the  coefficient  of  x  is  an  odd  number. 

The  second  is  the  shortest,  and  is  therefore  2i  favorite  with 
experts  in  algebra. 

The  advantage  of  the  third  is,  that  it  always  avoids 
fractions  in  completing  the  square. 


The  student  should  exercise  his  judgment  as  to  the  method 
best  adapted  to  his  purpose. 

Notes.  When  the  coefficient  of  x  is  an  even  number,  how  proceed  ?    Object  oJ 
multiplying  by  coefficient  Qix'^'i    By  4  ? 


182  AFFECTED     QUADRATICS. 

EXAMPLES. 

Find  the  yalue  of  x  in  the  following  equations: 

1.  x^  —  ^x^—i,  17.     3:^2  _  7;;c _  20  =  o. 

2.  a;2_^^_,_^^  jS.     ^x^ -- ido  ^  IX, 

3.  2a;2  __  y^  _.  _  2.  19.       2X^—  2X=\\. 

4.  a;2  -{-  loic  =  24.  20.     (ir  —  2)  (a;  —  i)  =  6. 

5.  6:^2— 130:  + 6  =  0.  21.     4(i»2__  i)  _4^__  I, 

6.  14a;  —  a;2  =  33.  22.     (2X  —  3)2  :=  8i?;. 

7.  iz;2— 3  =  — _^.  23.     3ic— 2  = 


6  •'^'     ^-      --a;-i 

8.     ^^ = ^-^.  24.     4a; =  14. 

16        100  —  ga?  „        t        40? 

to.     -  +  -  =  -.  26.     a;«  4-  -  =  -. 

X      a      a  22 

11.  a^  +  2/?za;  =  i^.  27.     x^  —  2nx  =  m^  —  7A 

12.  a^  +  f=ii.  .28.     9£.:=£)^£Z13«. 

'3-      ^2_6^-+y-^-3- 


14 


4^  a;  —  I 9a?  +  7 

14  —  ic         3ic    ~~      X     * 

15.  2A/rc2  —  4^  —  J  __  _  ^^ 

16.  V^^S  +  6  =  a;  +  5. 

338.  An  Equation  which  contains  but  two  powers  of  the 
unknown  quantity,  the  mdex  of  one  power  being  kvice  that 
of  the  other,  is  said  to  have  the  Quadratic  Form. 

The  indices  of  these  powers  may  be  either  integral  or 
fractional. 

Thus,  a^— a;2  =  12  ;  aj^^  +  ar*  =  A  ;  and  ^x  —  ^x  =  c,  are  equations 
of  the  quadratic  form. 

Note. — Equations  of  this  character  are  sometimes  called  trinomial 
equations. 

338.  When  has  an  equation  the  ouadratic  forinf  2fo(e.  What  are  such  equatioDB 
called? 


AFFECTED     QUADRATICS.  183 

339.  Equations  of  the  quadratic  form  may  be  solved  by 
the  rules  for  affected  quadratics. 

I.  Given  a^  —  2X^  =  8,  to  find  x, 

Solution  — By  the  problem,        ic*  —  2a;'  =  8 
Completing  square,  x^  —  2X^  +  i  =  g 

Extracting  square  root,  aj^  —  i  =  i  3 

Transposing,  a:^  =  4  or  —  2 

Extracting  square  root  again,  a;  =  ±  2,  or  ±  ^y/— 2 

.    2.  Given  0^  —  40^  =  ^2,  to  find  x. 

Solution. — By  the  problem,  afi  —  ^  =  32 

Completing  square,  afi  —  4iB^  +  4  =  36 
Extracting  square  root,  ar*  —  2  =  ±  6 

Transposing,  etc.,  a;'  =1  8  or  —4 

Extracting  cube  root,  .  a?  =  2  or  ^y^— 4 

3.  Given  x^"  —  4bx^  =  a,  to  find  x. 

Solution.— By  the  problem,     x^*  —  45a?*  =  a 

Completing  square,    a;-^'*  —  4hx^  +  45*  =  a  +  45* 


Extracting  square  root,  a;»  —  26  =  ±  ya+4ljr' 


Transposing,  a;"  =  2&  ±  ya+^ 

Extracting  the  Tith  root,  x  =  V  26  ±  ya+4b^ 

4.  Given  x^  +  S  =  6x^,  to  find  x. 

5.  Given  x*  —  2x^  =  3,  to  find  ar. 
%.  Given  afi  —  ju^  =  o,  to  find  x. 

x^      X        1 
/.  Given {-  -  =  -^,  to  find  ar. 

2  ^  4      32' 

8.  Given  v^  +  ^\/x  —  i,  to  find  x. 

9.  Given  4a;  +  4\/x  +2  =  7,  to  find  x. 

10.  Given      ^,    '  _   =  - — -=--,  to  find  «. 
4  +  V  a;  VaJ 


184  AFFECTED     QUADRATICS. 


PROBLEMS. 

1.  Find  two  numbers  such  that  their  sum  is  12  and  theii 
product  is  32. 

2.  A  gentleman  sold  a  picture  for  I24,  and  the  per  cent 
lost  was  expressed  by  the  cost  of  the  picture.    Find  the  cost 

Note. — Let  ic^tlie  cost. 

Then =  the  per  cent. 

100  ^ 

X 

We  now  have  x  —  x  y.  —  =  24,  to  find  the  value  of  05.   - 
100 

3.  The  sum  of  two  numbers  is  10  and  their  product  is  24. 
What  are  the  numbers  ? 

4.  A  person  bought  a  flock  of  sheep  for  |8o ;  if  he  had 
purchased  4  more  for  the  same  sum,  each  sheep  would  have 
cost  %i  less.  Find  the  number  of  sheep  and  the  price  of 
each. 

5.  Twice  the  square  of  a  certain  number  is  equal  to  65 
diminished  by  triple  the  number  itself.  Required  the 
number. 

6.  A  teacher  divides  144  oranges  equally  among  her 
scholars ;  if  there  had  been  2  more  pupils,  each  would  have 
received  one  orange  less.  Required  the  number  in  the 
school. 

7.  A  father  divides  $50  between  his  two  daughters,  in 
such  a  proportion  that  the  product  of  their  shares  is  $600. 
What  did  each  receive? 

8.  Find  two  numbers  whose  sum  is  100  and  their  product 
2400. 

9.  The  fence  enclosing  a  rectangular  field  is  128  rods 
long,  and  the  area  of  the  field  is  1008  square  rods.  What 
are  its  length  and  breadth  ? 

10.  A  colonel  arranges  his  regiment  of  1600  men  in  a 
solid  body,  so  that  each  rank  exceeds  the  file  by  60  soldiers. 
IIow  many  does  he  place  in  rank  and  file  ? 


AFFECTED     QUADRATICS.  185 

11.  A  drover  buys  a  number  of  lambs  for  $50  and  sells 
them  at  I5.50  each,  and  thus  gains  the  cost  of  one  lamb. 
^Required  the  number  of  lambs. 

12.  The  sura  of  two  numbers  is  4  and  the  sum  of  their 
reciprocals  is  i.    What  are  the  numbers  ? 

13.  The  sum  of  two  numbers  is  5  and  the  sum  of  their 
cubes  65.     What  are  the  numbers? 

14.  The  length  of  a  lot  is  i  yard  longer  than  the  width 
and  the  area  is  3  acres.    Find  the  length  of  the  sides. 

15.  A  and  B  start  together  for  a  place  300  miles  distant; 
A  goes  I  mile  an  hour  faster  than  B,  and  arrives  at  his 
journey's  end  10  hours  before  him.  Find  the  rate  per  hour 
at  which  each  travels. 

16.  A  and  B  distribute  $1200  each  among  a  certain 
number  of  persons.  A  relieves  40  persons  more  than  B,  and 
B' gives  to  each  person  $5  more  than  A.  Required  the 
number  relieved  by  each. 

17.  Divide  48  into  two  such  parts  that  their  product  may 
be  252. 

18.  Two  girls,  A  and  B,  bought  10  lemons  for  24  ceiL% 
each  spending  12  cents ;  A  paid  i  cent  more  apiece  than  B: 
how  many  lemons  did  each  buy  ? 

19.  Find  the  length  and  breadth  of  a  room  the  perimeter 
of  which  is  48  feet,  the  area  of  the  floor  being  as  many 
square  feet  as  35  times  the  difference  between  the  length 
and  breadth. 

20.  In  a  peach  orchard  of  180  trees  there  are  three  more 
In  a  row  than  there  are  rows.  How  many  rows  are  there, 
and  how  many  trees  in  each  ? 

21.  Find  the  number  consisting  of  two  digits  whose  sum 
is  7,  and  the  sum  of  their  squares  is  29. 

22.  The  expenses  of  a  picnic  amount  to  $10,  and  this  sum 
could  be  raised  if  each  person  in  the  party  should  give  30  cts. 
more  than  the  number  in  the  party.  How  many  compose 
the  party  ? 


186  AFFECTED     QUADKATICS. 

23.  Find  two  numbers  the  product  of  which  is  120,  and 
if  2  be  added  to  the  less  and  3  subtracted  from  the  greater, 
the  product  of  the  sum  and  remainder  will  also  be  120. 

24.  Divide  36  into  two  such  parts  that  their  product  shall 
be  80  times  their  difference. 

25.  The  sum  of  two  numbers  is  75  and  their  product  is 
to  the  sum  of  their  squares  as  2  to  5.     Find  the  numbers. 

26.  Divide  146  into  two  such  parts  that  the  difference  of 
their  square  roots  may  be  6. 

27.  The  fore- wheel  of  a  carriage  makes  sixty  revolutions 
more  than  the  hind-wheel  in  going  3600  feet ;  but  if  the 
circumference  of  each  wheel  were  increased  by  three  feet,  it 
would  make  only  forty  revolutions  more  than  the  hind- 
wheel  in  passing  over  the  same  distance.  What  is  the 
circumference  of  each  wheel  ? 

28.  Find  two  numbers  whose  difference  is  16  and  their 
product  $6. 

29.  What  two  numbers  are  those  whose  sum  is  i  J  and  the 
sum  of  their  reciprocals  si  ? 

30.  Find  two  numbers  whose  difference  is  15,  and  half 
their  product  is  equal  to  the  cube  of  the  less  number. 

31.  xi  lady  being  asked  her  age,  said,  If  you  add  the 
square  root  of  my  age  to  half  of  it,  and  subtract  12,  the 
remainder  is  nothing.     What  is  her  age  ? 

32.  The  perimeter  of  a  field  is  96  rods,  and  its  area  is 
equal  to  70  times  the  difference  of  its  length  and  breadth. 
What  are  its  dimensions  ? 

33.  The  product  of  the  ages  of  A  and  B  is  120  years.  If 
A  were  3  years  younger  and  B  2  years  older,  the  product  of 
their  ages  would  still  be  120.     How  old  is  each  ? 

34.  A  man  bought  80  pounds  of  pepper,  and  ^6  pounds 
of  saffron,  so  that  for  8  crowns  he  had  14  pounds  of  pepper 
more  than  of  saffron  for  26  crowns;  and  the  amount  he 
laid  out  was  188  crowns.  How  many  pounds  of  pepper  did 
he  buy  for  8  crowns  ? 


SIMULTAKEOUS     QUADRATICS.  187 


SIMULTANEOUS     QUADRATIC     EQUATIONS. 
TWO    UNKNOWN    QUANTITIES. 

340.  A  Moniogeneotis  Equation  is  one  in  which 
the  sum  of  the  exponents  of  the  unknown  quantities  is  the 
same  in  every  term  which  contains  them. 

Thus,  x'^—y'^  =  7,  and  x^—xy  +  y^  =  13,  are  each  homogeneous. 

341.  A  Symmetrical  Equation  is  one  in  which 
the  unknown  quantities  are  involved  to  the  same  degree. 

Thus,  x''+y^  =  34,  and  x^y—xy^  =  34,  are  each  symmetrical. 

342.  Simultaneous  Quadratic  Equations  con- 
taining two  unknown  quantities,  in  general  involve  the 
principles  of  Biquadratic  equations,  which  belong  to  the 
higher  departments  of  Algebra. 

There  are  three  classes  of  examples,  however,  which  may 
be  solved  by  the  rules  of  quadratics. 

ist.  When  one  equation  is  quadratic,  and  the  other  simple, 
2d.  When  both  equations  are  quadratic  and  homogeneous, 
3d.  When  each  equation  is  symmetrical. 

343.  To    Solve    Simultaneous    Equations   consisting   of, a 

Quadratic  and  a  Simple  Equation. 

I.  Given  x^  -{-  y'^=z  13,  and  x  -{-  y  =z  5,  to  find  x  and  y. 


Solution.— By  the  problem,                  x^+y^  =  13 

(i) 

x+y=    5     . 

(2) 

By  transposition,                                         x  =  s—y 

(3) 

Squaring  each  side  of  (3)  (Art.  102),        x'^  =  2S  —  ioy+y^ 

(4) 

Substituting  (4)  in  (i),       2S—ioy  +  y'^+y'^  —  13 

(5) 

Uniting  and  transposing,               2y'^—ioy  =  — 12 

(6) 

Comp.  sq.  (Art.  336,710(6),      4y^—2oy  +  2S  =  —24  +  25 

(7) 

Extracting  root,                                     2^—5  =  ±  i 

.-.    y=:3  or  2. 

Substituting  value  of  y  in  (3),  a;  =  2  or  3.    Hence,  the 

340.  What  is  a  homogeneoua  equation  ?    341.  A  symmetrical  equation  ? 


188     '  SIMULTAN^EOUS     QUADRATICS. 

Rule. — Find  the  value  of  one  of  the  unhnoivn  quantities 
in  the  simple  equation  ly  transposition,  and  substitute  this 
value  in  the  quadratic  equation.     (Arts.  221,  223.) 

^olye  the  following  equations: 

2.       0:2+^2:^25,  5.       a:2  +  ^2:=244, 

y  —X  =     2. 
6.     3^:2  _  ^2 —25 1, 

^  +  Ay  =  38. 

4.     a^J_^2_28,  7.     8a;2_j_  ^^2_  728, 

6y  —  x=z  15. 

344.    To   Solve   Simultaneous    Equations  which    are    both 
Quadratic  and  Homogeneous. 

8.  Given  x^-\-xy  =  40,  and  y'^-^xy  =  24,  to  find  x  and  y. 


X  +?/  -- 

=    7- 

x^-^-f-- 

=  74, 

x  +  y=i 

:  12. 

x^-f-. 

=  28, 

x-y  = 

:  2. 

jUTIon. — By  the  problem. 

x^  +  a^  =  4o 

(I) 

»<                  «c 

y^  +  xy  =  24 

(2) 

Let 

x  =  py 

(3) 

Substituting  ^3^  in  (i). 

pY  +pf  -  40 

(4) 

"     (2), 

/+i>2/^  =  24 

(5) 

Factoring,  etc.,  (4), 

^      p'+p 

(6) 

*'    (5), 

^     i+P 

(7) 

Equating  (6)  and  (7), 

40        24 
p'+p     i+p 

(8) 

Clearing  of  fractions. 

S  +  5P  =  3P'  +  3P 

(9) 

Transposing,  etc., 

3P'-2p  =  5 

(ID) 

Comp.  sq.,  3d  meth.  (Art.  336), 

gp'^—6p+i  =  15  +  1  =  16 

(II) 

Extracting  root, 

3P-1  =  ±  4 

(12) 

Transposing,  3^  =  i  ±  4 

Dropping  the  negative  value,  p  =  | 

Substituting  value  of  ^  in  (7),  y^  =  24-^(1  +|)=9 
Extracting  root,  y  =  ±  3 

Substituting  value  of  p  and  y  in  (3),  a?  =  fx  ±3  =  ±5. 

Hence,  the 

343.  Rule  for  solution  of  equations  coneieting  of  a  quadratic  and  simple  equ^ 
tion? 


SIMULTANEOUS     QUADRATICS.  189 

Rule. — I,  For  one  of  the  unknown  quantities  substitute 
the  product  of  the  other  into  an  auxiliary  quantity,  and  then 
find  the  value  of  this  auxiliary  quantity. 

II.  Find  the  values  of  the  unknown  quantities  by  substi- 
tuting the  value  of  the  auxiliary  quantity  in  one  of  the 
equations  least  involved. 

Note. — ^An  auxiliary  quantity  is  one  introduced  to  aid  in  the  sola 
>,ion  of  d  problem,  as  p  in  the  above  operation. 


9.  Given 
A.nd 


It^^""    o^' 1  to  find  a;  and  y. 

10.  Given  x  ^y  =     2,  ].«,         , 

1 1.  Gireu  3^  -  72/'  -  -  I,  I  t„  g„3  ^  ,„^ 
And  4^y=     24,  )  ^ 

1 2.  Given  ^  -  ^y  +  3^  =  '9, 1  to  fi^a  ^  ^„3 
And  xy=i5,)  ^ 

345.  To  Solve  Simultaneous  Quadratic  Equations  when  each 
Equation  is  Symmetrical. 

13.  Given  x  +  y  =zg,  and  xy  =  20,  to  find  x  and  y. 


.UTiON.— By  the  problem,              x+y=   9 

(I) 

xy  =  20 

(2) 

Squaring  (i),                        aj»  +  2a'2^ + /  =  8 1 

(3) 

Multiplying  (2)  by  4.                        40;^  =  80 

(4) 

Subtracting  (4)  from  (3),     a?—2xy+y^  -    I 

(5) 

Extracting  sq.  root  of  (5),                x—y  —  ±1 

(6) 

Bringing  down  (i),                           x  +  p  =      9 

Adding  (i)  and  (6),                        2X       =    10  or  8 

(7) 

Removing  coefficient,                            a;  =      5  or  4 

Substituting  value  of  a;  in  (i),             y  =     4  or  5 

Notes. — i.  The  values  of  x  and  y  in  these  equations  are  not  equal, 
but  interchangeable ;  thus,  when  x  =  s,  y=4  ;  and  when  aj  =  4,  y  =  5 

344.  How  solve  equations  which  are  both  quadratic  and  homogeneous  ?     I^ote, 
What  is  an  auxiliary  quantity  ? 


190  SIMULTANEOUS     QUADRATICS. 

2.  The  solution  of  this  class  of  problems  'caries  according  to  the 
given  equations.  Consequently,  no  specific  rules  can  be  givren  that 
will  meet  every  case.  But  judgment  and  practice  will  readily  supply 
expedients.    Thus, 

I.  When  the  sum  and  product  are  given.     (Ex.  13,  15.) 
Find  the  difference  and  combine  it  with  the  sum,    (Art.  2  24.) 

II.  When  the  difference  and  product  are  given.   (Ex.  1 6.) 
Find  the  sum  and  combine  it  ivith  the  difference, 

III.  When  the  sum  and  difference  of  the  same  powers  are 
given.     (Ex.  14,  17.) 

Combine  the  two  equations  by  addition  a7id  subtraction, 

rV*.  When  the  members  of  one  equation  are  multiples  of 
the  other.    (Ex.  18.) 

Divide  o?ie  by  the  other,  and  then  reduce  the  resulting 
equation, 

14.  Given  a;^  -f  ^^  =  5,     (i)  )  .    «    ,  , 

^    ^    ,       1      ^1  X  ^  to  find  a;  and  V. 

And     x»  —y^  =.  i,     (2)  ) 

Solution, — Adding  (i)  and  (2),  and  dividing,  a;^  =    3 

Involving,  a?   =  27 

Subtracting  (2)  from  (i),  etc  y*  =    2 

Involving,  y   =    8 

15.  Given  2^  +  ^=27,  ],«,         , 

^     .    ,         ^^        „"  ^  to  findrraudy. 
And  xy  =  180,  \  ^ 

16.  Given  a?  —  v=    14,  )./>;,         , 

.    ,  ^  >  to  find  X  and  y. 

And  xy  =  147,  ) 

,"*"'\'^^'ltofinda:andy. 
*  —  ^^  =  3?  ' 


17.  Given  x^  +  y^  =  7, 
And    X 


18.  Given  o^y^  +  x^f  =z  12,]  ,    ^  ^         , 
A    :i      00        ^  r  to  fin<i  ^  and  y. 
And    a?y  -{- xy^  =   6,  )  ^ 


Note,— What  is  true  of  the  solution  of  Bimultaneous  quadratics?  When  the  tmm 
and  product  are  ffiven,how  proceed?  When  the  difference  and  product?  When 
the  sum  and  difference  of  the  same  powers  are  given  ?  When  the  members  of  one 
tquation  are  multiples  of  the  other  ? 


SIMULTANEOUS     QUADBATIC8.  193 


PROBLEMS. 

1.  The  difference  of  two  numbers  is  4,  and  the  difference 
of  their  cubes  448.     What  are  the  numbers  ? 

2.  A  man  is  one  year  older  than  his  wife,  and  the  product 
of  their  respective  ages  is  930.     What  is  the  age  of  each  ? 

3.  Required  two  numbers  whose  sum  multiplied  by  the 
greater  is  180,  and  whose  difference  multiplied  by  the  less 
is  16. 

4.  In  an  orchard  of  1000  trees,  the  number  of  rows 
exceeds  the  number  of  trees  in  each  row  by  15.  Required 
the  number  of  rows  and  the  number  of  trees  in  each  row. 

5.  The  area  of  a  rectangular  garden  is  960  square  yards, 
and  the  length  exceeds  the  breadth  by  16  yards.  Required 
the  dimensions. 

6.  Subtract  the  sum  of  two  numbers  from  the  sum  of 
their  squares,  and  the  remainder  is  78 ;  the  product  of  the 
numbers  increased  by  their  sum  is  39.  What  are  the 
numbers  ? 

7.  Find  two  numbers  whose  sum  added  to  the  sum  of 
their  squares  is  188,  and  whose  product  is  77. 

8.  A  surveyor  lays  out  a  piece  of  land  in  a  rectangular 
form,  so  that  its  perimeter  is  100  rods,  and  its  area  589 
square  rods.     Find  the  length  and  breadth. 

9.  Required  two  numbers  whose  product  is  28,  and  the 
sum  of  their  squares  65. 

10.  A  regiment  of  soldiers  consisting  of  1154  men  is 
formed  into  two  squares,  one  of  which  has  2  more  men  on  a 
side  than  the  other.  How  many  men  are  on  a  side  of  each 
of  the  squares  ? 

11.  Required  two  numbers  whose  product  is  3  times  their 
sum,  and  the  sum  of  their  squares  1 60. 

12.  What  two  numbers  are  those  whose  product  is  6  times 
their  difference,  and  the  sum  of  their  squares  13  ? 

CSee  A'^r)cnclix,  p.  290.) 


CHAPTER    XVII. 
RATIO    AND    PROPORTION. 

346.  Ratio  is  the  relation  which  one  quantity  bearb  k 
pother  with  respect  to  magnitude. 

347.  The  Ter^ns  of  a  Ratio  are  the  quantities 
compared.  The  first  is  called  the  Antecedent,  the  second 
the  Consequent y^  and  the  two  together,  a  Couplet 

348.  The  HUjn  of  ratio  is  a  colon  :  \  placed  between  the 
two  quantities  compared. 

Ratio  is  also  denoted  by  placing  the  consequent  under  the 
antecedent,  in  the  form  of  2^  fraction. 

Thus,  the  ratio  of  a  to  6  is  written,  a  :  &,  or  ^  • 

349.  The  Measure  or  Value  of  a  ratio  is  the  quotient 
of  the  antecedent  divided  by  the  consequent,  and  is  equal 
to  the  value  of  the  fraction  by  which  it  is  expressed. 

Thus,  the  measure  or  value  of  8  :  4  is  8-7-4  =  2. 

Note. — That  quantities  may  have  a  ratio  to  each  other,  they  must 
be  so  far  of  the  same  nature,  that  one  can  properly  be  said  to  be  equal 
to,  or  greater t  or  les8  than  the  other. 

Thus,  a  foot  has  a  ratio  to  a  yard,  but  not  to  an  hour,  or  a  pound 

350.  A  Simple  Itatio  is  one  which  has  but  two  terms  *, 

fts,     a\h,    8 : 4. 

346.  What  is  ratio  ?  347.  What  are  the  terms  of  a  ratio  ?  348.  The  pign  ?  How 
»lso  is  ratio  denoted  ?  349.  The  measure  or  value  ?  Note.  What  quantities  have  a 
ratio  to  each  other  ?    350.  What  is  a  simple  ratio  ? 

*  Ardecedent,  Latin  ante,  before,  and  cedere,  to  go,  to  preceoe. 
Consequent,  Latin  con,  and  sequi,  to  follow. 

f  Tlie  sign  of  ratio  :  is  derived  from  the  sign  of  division  -f-,  the 
horizontal  line  being  dropped. 


RATIO.  193 

351.  A  Compound  Matio  is  the  product  of  two  or 
more  siinple  ratios. 

Thus,     4  •  2  [  are  eacli  simple  But  4x9:  2x3 

9:3$         ratios.  is  a  compound  ratio. 

Note. — The  nature  of  compound  ratios  is  the  same  as  that  of  sim- 
ple ratios.  They  are  so  called  to  denote  their  origin,  and  are  usually- 
•expressed  by  writing  the  corresponding  terms  of  the  simple  ratios  one 
•mder  another,  as  above. 

352.  A  Direct  Ratio  arises  from  dividing  the  ante- 
cedent by  the  consequent. 

353.  An  Inverse^  or  Heciprocal  Matio  arises 
from  dividing  the  consequent  by  the  antecedent,  and  is  the 
same  as  the  ratio  of  the  reciprocals  of  the  two  numbers 
compared. 

Thus,  the  direct  ratio  of  a  to  a&  =  -r,  or  v,  and  that  of  4  to 

4  I 

12  =  — ,  or  -  • 
12  3  . 

The  inverse  ratio  of  a  to  a&  =  — ,  or  6 ;  of  4  to  12  =  — ,  or  3. 

tt  4 

It  is  the  same  as  the  ratio  of  the  reciprocals,  -  to  -^ ,  and  -  to  —  • 

a      ab  4      12 

Note. — A  reciprocal  ratio  is  expressed  by  inverting  the  fraction 
which  expresses  the  direct  ratio.  When  the  colon  is  used,  it  is 
expressed  by  inverting  the  m^der  of  the  terms. 

354.  The  ratio  between  two  fractions  which  have  a 
common  denominator,  is  the  same  as  the  ratio  of  their 
numerators. 

Thus,  the  ratio  of  f  :  f  is  the  same  as  6:3. 

Note. — When  the  fractions  have  different  denominators,  reduce 
them  to  a  common  denominator;  then  compare  their  numerators. 
(Art.  175.) 

355.  A  Ratio  of  Equality  is  one  in  which  the  quan- 
tities compared  are  equal,  and  its  value  is  a  unit  or  i. 

351.  What  is  a  compound  ratio?  l<!ote.  Why  so  called?  352.  What  is  a  direct 
ratio  ?    353.  A  reciprocal  ?    355.  What  is  a  ratio  of  equality  ? 

*  Inverse,  from  the  Latin  in  and  verto,  to  turn  upside  down,  to  invert. 

9 


194  RATIO. 

356.  A  !Ratio  of  Greater  Inequality  is  one  whose 
antecedent  is  greater  than  its  consequent,  and  its  value  is 
greater  than  i. 

357.  A  IRatio  of  Less  Inequality  is  one  whose 
antecedent  is  less  than  its  consequent,  and  its  value  is  less 
than  I. 

358.  A  Duplicate  Hatio  is  the  square  of  a  simple 
ratio.  It  arises  from  multiplying  a  simple  ratio  into  itself, 
or  into  another  equal  ratio. 

359.  A  Triplicate  Matio  is  the  cule  of  a  simple  ratio, 
and  is  the  product  of  three  equal  ratios. 

Thus,  the  duplicate  ratio  of  a  to  5  is  a* :  6*. 
The  triplicate  ratio  of  a  to  &  is  a^  :  5^. 

360.  A  Suhduplicate  Hatio  is  the  square  root  of  a 
simple  ratio. 

361.  A  Subtriplicate  Matio  is  the  cuhe  root  of  a 
simple  ratio. 

Thus,  the  svbduplicate  ratio  of  a;  to  y  is  ^\/x  :  ^/y. 
The  sicbtriplicate  ratio  of  a;  to  y  is  y^x  :  ^y,  etc. 

362.  Since  ratio  may  be  expressed  in  the  form  of  a 
fraction,  it  follows  that  changes  made  in  its  terms  have  the 
same  effect  on  its  value,  as  like  cUamjes  in  the  terms  of  a 
fraction.     (Art.  167.)     Hence, the  following 

PRINCIPLES. 

I®.  Multiplying  the  antecedent,  or  )  ,^  ...  ,.     ^- 

T^'  'f'     J.1  ^  \  MuUivhes  tho  ratio. 

Dividing  the  consequent,  j 

2**.  Dividing  the  antecedent,  or      )  r.-  -t     ^t 

!•#- 7,.  7  .      ,1  ±      \  Divides  the  ratio. 

Multiplying  the  consequent,      ) 

3**.  Multiplying  or  dividing  loth  \  Does  7iot  alter  the  value 

terms  hy  the  same  quantity,  f      of  the  ratio. 

356.  Of  greater  inequality?  357.  Of  lef?  inequality?  358.  A  duplicate  ratio? 
359.  Triplicate?  360.  Subduplicate ?  361.  Subtriplicate?  363.  Name  Principle  i. 
Principle  a     Principle  a> 


EATia  Idd 


EXAMPLES. 


1.  What  is  the  ratio  of  4  yards  to  4  feet? 

Solution.  4  yards  =  12  feet ;  and  the  ratio  of  12  ft.  to  4  ft.  ia  y 

2.  What  is  the  ratio  of  6a^  to  2X  ?  Ans,  3a:, 

3.  What  is  the  ratio  of  40  square  rods  to  an  acre  ? 

4.  What  is  the  ratio  of  i  pint  to  a  gallon  ? 

5.  What  is  the  ratio  of  64  rods  to  a  mile  ? 

6.  What  is  the  ratio  of  Sa^  to  4a  ? 

7.  What  is  the  ratio  of  isaic  to  ^ab? 

8.  What  is  the  ratio  of  $5  to  50  cents  ? 

9.  What  is  the  ratio  of  75  cents  to  16  ? 

10.  What  is  the  ratio  of  35  quarts  to  35  gallons  P 

11.  What  is  the  ratio  of  20,^  to  4a  ? 

12.  What  is  the  ratio  otx^  —  y^tox-^-r/? 

13.  What  is  the  compound  ratio  of  9:12  and  8  :  15  ? 
Solution.    9  x  8  =  72,  and  12  x  15= i8a    Now  72-5-180  =  t^,  Ana 
Or,  9  :  12  =  ■^,  and  8  :  15  =  ^V    Now  j\  x  yV  =  t¥o  =  tu»  ^^■ 

14.  What  is  the  compound  ratio  of  8  :  15  and  25  :  30? 

15.  What  is  the  compound  ratio  of  a:b  and  2h  :  ^ax? 

16.  Eeduce  the  ratio  of  9  to  45  to  the  lowest  terms. 
Solution.    9  :  45  =  t\»  and  ^\  =  |,  Ans. 

17.  Reduce  the  ratio  of  24  to  96  to  the  lowest  terms. 

18.  Reduce  the  ratio  of  144  to  1728  to  the  lowest  terms. 

19.  What  kind  of  ratio  is  25  to  25  ? 

20.  What  kind  of  a  ratio  is  ab:  ab? 

21.  What  kind  of  ratio  is  35  to  7  ? 

22.  What  kind  of  ratio  is  6  to  48  ? 

23.  Which  is  the  grearter,  the  ratio  of  15  :  9,  or  38  :  19? 

24.  Which  is  the  greater,  the  ratio  of  8  :  25,  or  V4  '-  V25, 

25.  If  the  antecedent  of  a  couplet  is  56,  and  the  ratio  8 
what  is  the  consequent? 

26.  If  the  consequent  of  a  couplet  is  7,  and  the  ratio  14, 
what  is  the  antecedent  ? 


196  pBOPOBiioifir. 


PROPORTION. 

363.  Proportion  is  an  equality  of  ratios. 

Thus,  the  ratio  8  ;  4  =  6  :  3,  is  a  proportion.    That  is. 
Four  quantities  are  in  proportion,  when  the  first  is  the  same  mvlti- 
pie  or  part  of  the  second  that  the  third  is  of  the  fourth. 

364.  The  Sign  of  Proportion  is  a  double  colon  :  :,♦ 
or  the  sign  =.     Thus, 

The  equality  between  the  ratio  of  a  to  5  and  c  to  (?  is  expressed  by 

a  :  6  : :  c :  <f  ,  or  by  T  =  3 
0     d 

The  former  is  read,  "  a  is  to  &  as  c  is  to  (f ;"  the  latter,  *'&  is  contained 

In  a  as  many  times  as  d  is  contained  in  c.** 

Note.  —Each  ratio  is  called  a  couplet ,  and  each  term  a  proportional. 

365.  The  Terms  of  a  proportion  are  the  quantities 
compared.  ^\iq  first  and  fourth  are  called  the  extremes,  the 
second  and  third  the  means, 

366.  In  every  proportion  there  must  be  at  least  four 
terms  ;  for  the  equality  is  between  two  or  more  ratios,  and 
each  ratio  has  two  terms. 

367.  A  proportion  may,  however,  be  formed  from  three 
quantities,  for  one  of  the  quantities  may  be  repeatedy  so  as 
to  form  two  terms ;  as,  a\h  : :  J :  c. 

Note. — Care  should  be  taken  not  to  confound  propoHion  with  ratio. 
In  common  discourse,  these  terms  are  often  used  indiscriminately. 
Thus,  it  is  said,  "  The  income  of  one  man  bears  a  greater  proportion 
to  his  capital  than  that  of  another,'*  etc.  But  these  are  loose  expressions 

In  a  simple  ratio  there  are  but  two  terms,  an  antecedent  and  a 
cjonsequent ;  whereas,  in  a  proportion  there  must  at  least  be  four 
terms.    (Arts.  350,  366.) 

363.  What  is  proportion?  364.  The  sign  of  proportion?  Note.  What  is  each 
ratio  called  ?  365.  What  are  the  terms  of  a  proportion  ?  366.  How  many  terms  in 
every  proportion  ?    367.  How  form  a  proportion  from  three  quantities  ? 

*  The  sign  : :  is   derived  from  the  sign  of  equality  =,  the  four 

points  being  the  terminations  of  the  lines. 


PBOPOETION.  197 

Again,  one  ratio  may  be  greater  or  less  than  another,  but  one 
proportion  is  neither  greater  nor  less  than  another.  For  equality  does 
not  admit  of  degrees.  In  scientific  investigations,  this  distinction 
should  be  carefully  observed. 

368.  A  Mean  Proportional  between  two  quantities 
is  the  middle  term  or  quantity  repeated,  in  a  proportion 
formed  from  three  quantities. 

369.  A  Third  Projjortional  is  the  last  term  of  a 
proportion  having  three  quantities. 

Thus,  in  the  proportion  a:h  ::  b:e,b\ask mean  proportional,  and 
e  a  third  proportional. 

370.  A  Direct  Proportion  is  an  equality  between 

two  direct  ratios ;  as,  a:b  : :  c:d,  $16  : :  4:8. 

371.  An  Inverse  or  Heciprocal  Proportion  is  an 

equality  between  a  direct  and  reciprocal  ratio ;  as, 
8:4  ::  i'.h 

372.  Analogous  Terms  are  the  antecedent  and  con- 
sequent of  the  same  couplet. 

373.  Homologous  Terms  are  either  two  antecedents 

or  two  consequents. 

PROPOSITIONS. 

374.  A  Proposition  is  the  statement  of  a  truth  to  be 
proved,  or  of  an  operation  to  be  performed. 

Propositions  are  of  two  kinds,  theorems  and  proUems, 

375.  A  TJieorem  is  something  to  be  proved. 

376.  A  Problem  is  something  to  be  done. 

377.  A  Corollary  is  a  principle  inferred  from  d 
preceding  proposition. 

368.  What  is  a  mean  proportional  ?  369.  Wliat  is  a  third  proportional  ?  370.  A 
direct  proportion?  371.  An  inverse  or  reciprocal  proportion?  372.  What  ar« 
analop:ou8  terms  ?  373.  Homologoni??  374.  What  is  a  proposition  ?  How  divided? 
375.  What  is  a  theorem ?    376.  A  problem?    377.  A  .^roUary? 


198  PEOPORTIOK, 

378.  Tho  more  important  theorems  in  proportion  are  the 
following: 

Theorem  L 

If  four  quantities  are  'proportional^  the  product  of  the 
extremes  is  equal  to  the  product  of  the  means. 

Left  a  :  d  ::  e  :  d 

By  An.  363,  |=| 

dearing  of  fractions,  a<2  =  && 

Verification  by  NuMBERa 
Given,    a:  4::  8:  16;    and    2x16  =  4x8. 

Cor. — ^The  relation  of  the  four  terms  of  a  proportion  to 
each  other  is  such,  that  if  any  three  of  them  are  given,  the 

fourth  may  be  found. 

Thus,  since  ad  —  be,  it  follows  that 

a  =  bc+dt  b  =  ad-T-c,  e  =  ad-r-b,  and  d  =  bc-t-a,    (Ax.  5.) 

Notes. — i.  The  rule  of  Simple  Proportion  in  Arithmetic  is  founded 
upon  this  principle,  and  its  operations  are  easily  proved  by  it. 

2.  This  theorem  furnishes  a  very  simple  test  for  determining 
whether  any  four  quantities  are  proportional.  W©  have  only  to 
multiply  the  extremes  together,  and  the  means. 


Theorem  IL 

If  three  quantifies  are  proportional,  the  product  of  the 
extremes  is  equal  to  the  square  of  the  mean. 

Let  a  :  6  : :  b  •  t 
By  Art.  363,  |=:- 

Oearing  of  fractions,  ac  =  b*.    . 

Again,    9:6::  6:4,    and  4x9  =  6'. 

Cor. — A  mean  proportional  between  two  quantities  is 
equal  to  the  square  root  of  their  product 


PEOPOETIO]Sr.  199 


Theorem  HL 

ff  the  product  of  two  quantities  is  equal  to  the  product  of 
two  others,  the  four  quantities  are  proportional ;  the  factors 
of  either  product  being  taken  for  the  extremes,  and  the  factors 
9fthe  other  for  the  means. 

Let  adr=lfe 

DividingbjM,  1  =  1 

Or,  by  Art.  363,  a:b  ::  e  :  d, 

Agoia,   4x6  =  3x8,    and   4:3::  8  :& 


Theoeem  IV. 

If  four  quantities  are  proportional,  they  are  proportional 
when  the  means  are  inverted. 

Let    aih  :i  ei  d,   then   a  i  e  11  h:  d 

a     e 


For,  hj  Art.  363, 


h~d 


Multiplying  by  -.  *  s=  j 

Or,  aie  II  hi  d. 

Again.    3  :  6  :  •  4  :  8,    and    3  :  4  : :  6  :  8.    (Th.  i.) 

Note.  —  This    change   in    the   order  of  the   means   \a  called 
**  AlterTuUian,** 


Theorem  V. 

If  four  quantities  are  proportional,  they  are  proportional 
when  the  terms  of  each  couplet  are  inverted. 

Le.    a  ih  M  e  \  dt    then    h  ',  a  .:  d  \  t 
By  Theorem  i,  '       ad  =  he 

By  Theorem  3,  h  :  a  ::  d  :  e. 

Again,    6  :  2  ::  15  :  5,    then    2  :  6  ::  5  :  15.    (Th.  i.) 

Cor. — ^If  the  extremes  are  inverted,  or  the  order  of  the 
terms,  the  quantities  will  be  proportional. 


200  PROPORTIOK. 

Notes. — i.  If  the  terms  of  onlj  one  of  the  couplets  are  inverted, 
ihe  proportion  becomes  reciprocal, 

2.  The  change  in  the  order  of  the  terms  of  each  couplet  Ls  called 
^Inversion** 

3.  This  proposition  supposes  the  quantities  compared  to  be  of  the 
same  kind.    Thus,  a  line  has  no  relation  to  weight.  (Art  349,  note.) 

Theorem  VL 

If  four  quantities  are  proportional ,  ttvo  analogous  or  twt^ 
homologous  terms  may  be  multiplied  or  divided  by  the  sam4 
quantity  without  destroying  the  proportion. 

Let  a:h     ::  e    I  d 

Multiplying  analogous  terms,    am  :  bm  ::  e     :  d 
and  a    :  &     11  em  i  dm 

^'  h^d 

Hence.  (Art  362.  Prin.  3).       ^  =  |.    and    |  =  ^ 

Multiplying  homologous  terms,    am  :  5  : :  cm  i  d 

And  a  '.  hm  '.:  c  :  dm. 

.„.         , .  ^     am     em         3     a        e 

Hence.  (Ax  4. 5),    X=T'    ""^    i^=^ 

Dividing  analogous  terms,  —  :  —  :  r  e  :  d, 

and  «  :  6  ;  •  —  :  — 

m     m 

Dividing  homologous  terms,         —  :  b  ::  —  :  d 

b  d 

and  ^^  •  ;r  • '  ^  •  ;r 

m  m 

Clearing  of  fractions  (Th.  i),  ad=:be 

Cor. — All  the  terms  of  a  proportion  maybe  multiplied 
01  divided  by  the  same  quantity  without  destroying  the 
proportion. 

Notes. — i.  When  the  homologous  terms  are  multiplied  or  divided, 
both  ratios  are  equally  increased  or  diminished. 

2.  When  the  analogoiLs  terms  are  multiplied  or  divided,  the  ratios 
are  not  altered. 


PEOPOHTION.  201 


Theokem  VIL 

If  four  quantities  are  proportional,  the  sum  of  the  first 
and  second  is  to  the  second,  as  the  sum  of  the  third  and  fourth 
is  to  the  fourth. 

Let    a    h  II  ei  d,    then    fl+6  :  6  ::  e-\-d  :  d 
_  a     c 

*^°''  »=5 

Adding  I  to  each  member,         ^  +  i  =  -%  +  1.    (Ax.  a.) 

Incorporating  i,  — r—  =  —-^ 

Therefore  (Art.  363),  a+h  il  ::  c  +  d  :  d 

Again,    4:2  ::  6:3,    then    4+2  :  2  ::  6  +  3  :  3 
KoTE. — This  combination  is  sometimes  called  **  Composition.'* 


Theorem  VIII. 

//  four  quantities  are  proportional,  the  difference  of  the 
first  and  the  second  is  to  the  second,  as  the  difference  of  the 
third  and  fourth  is  to  the  fourth. 

Let    a  :  6  : :  c  :  d,   then    a— &  :  h  ::  c—d  :  d 

Subtracting  i  from  each  member,    r-  —  i  =  -  —  x 

0  d 

-  ,.  a— 6     c—d 

Incorporating  —1,  — v—  =  ^      - 

Therefore,  a— 6  :  6  : :  c—d  :  <2 

Again,     4:2  : :  6:3,    then    4—2  12::  6—3  :  3 

Note. — This  comparison  is  sometimes  called  *' Division" * 

*  The  technical  terms.  Composition  and  Division,  are  calculated 
rather  to  perplex  than  to  aid  the  learner,  and  are  properly  falling  into 
disuse.  The  objection  to  the  former  is,  that  it  is  liable  to  be  mistaken 
for  the  composition    or  compounding   of   ratios,  whereas    the    two 


202 


PEOPORTION. 


Theoeem  IX. 

If  two  ratios  are  respectively  equal  to  a  third,  they  are 
equal  to  each  other. 


Let    a  :  b  ::  m  :  n, 
a  _m 
l~"n' 


Then 


and 
and 


d  ::  m  I  n 
d  ~"  » 


That  is,    a  :  b  =  e  :  d 


Again,    12  :  4  =  6  :  2,    and       9:3  =  6:3 
A    12  :  4  =  9  :  3 


Theoeem  X. 

When  any  numler  of  quantities  are  proportional,  any 
antecedent  is  to  its  consequent,  as  the  sum  of  all  the  ante- 
cedents is  to  the  sum  of  all  the  consequents. 

Let  a  I  b  ::  e  :  d  ::e:/,  etc. 

Then  a  :  b  :i  a+c+e  :  b+d+f,  etc. 

For  (Th.  i),  ad  — be 

And,     **  af  —  be 

Also,  ab  =  ba 

Adding  (Ax,  2), 
Factoring, 


ab+ad+af  =  ba+bc+be 
a(b+d+f)  =  b{a+c+e) 


Hence,  (Th.  3),    a  :b  i:  {a+e+e,  etc.)  :  (J)+di-f,  etc.) 

» ■         — — — — — — — — — - — ■ — ~— 

operations  are  entirely  different.     In  one  the  terms  are  added,  in  the 
other  they  are  multiplied  together.    (Art.  351,) 

The  objection  to  the  latter  is,  that  the  change  to  which  the  term 
division  is  here  applied,  is  effected  by  subtraction,  and  has  no 
reference  to  division,  in  the  sense  the  word  is  used  in  Arithmetic  and 
Algebra.  Moreover,  the  alteration  in  the  terms  of  Theorem  6  is 
produced  by  actual  division.  Usage,  however  ancient,  can  no  longer 
justify  the  employment  of  the  same  word  in  two  different  senses,  in 
explaining  the  same  subject. 


PEOPOETI02!r.  203 


Theorem  XL 

If  the  corresponding  terms  of  two  or  more  proportions  are 
multiplied  together,  the  products  will  he  proportional. 

Let  a  ih  ::  ci  d,    and    e  i  f  i:  g  :  h 

Then  ae  :hf  ::  eg  :  dh 

For.  -=5  '«'<1        7=f 

Halt  ratios  together  (Ax.  4),  h^~^ 

Hence,  (Th.  3),       ae  :  bf  z:  eg  :  dh. 


Theoeem  xn. 

If  four  quantities  are  proportional,  like  powers  or  roots 
of  these  quantities  are  proportional. 

Let  a  :  b  : :  e  :  d,       then       t  =  3 

0     a 

■DA  a«     c* 

Hence  (Th.  3),  «»  :  6»  ; ;  c»  :  d» 

Extracting  sq.  root,        a*  :  6*  : :  c*  :  d* 
Again,     2:3  : :  4:6,    then    2«  :  3'  : :  4"  :  6* 

In  like  manner,        /y^  :  ^/g  : :   -y/iS  :  -y/is. 

Note.— The  index  »  may  be  either  integral  or  fractional. 


Theorem  XIIL 

Equimultiples  of  two  quantities  are  proportional  to  the 
ruantities  themselves. 

Smce         1=3,   hy  Art.  362,  Pnn.  3,    r—  =  ^ 
fid"  '^     bm      d 

Hence,  am  :  im  ii  c  :  d. 


204  PEOPORTIOK 


PROBLEMS. 

1.  The  first  three  terms  of  a  proportion  are  6,  8,  and  3. 
What  is  the  fourth  ? 

Let  x  =  the  fourth  term. 

Then  6:8  : :  3  :  x 

/.    6x  =  24,        and        x  =  4. 

2.  The  last  three  terms  of  a  proportion  are  8,  6,  and  12. 
What  is  the  first? 

3.  Eequired  a  third  proportional  to  25  and  400. 

4.  Required  a  mean  proportional  between  9  and  16. 

5.  Find  two  numbers,  the  greater  of  which  shall  be  to 
the  less,  as  their  sum  to  42  ;  and  as  their  difference  to  6. 

6.  Divide  the  number  18  into  two  such  parts,  that  the 
squares  of  those  parts  may  be  in  the  ratio  of  25  to  16. 

7.  Divide  the  number  28  into  two  such  parts,  that  the 
quotient  of  the  greater  divided  by  the  less  shall  be  to  the 
quotient  of  the  less  divided  by  the  greater  as  32  to  18. 

8.  What  two  numbers  are  those  whose  product  is  24,  and 
the  difference  of  their  cubes  is  to  the  cube  of  their  difference 
as  19  to  I  ? 

9.  Find  two  numbers  whose  sum  is  to  their  difference  as 
9  is  to  6,  and  whose  difference  is  to  their  product  as  i  to  12. 

10.  A  rectangular  farm  contains  860  acres,  and  its  length 
is  to  its  breadth  as  43  to  32.  What  are  the  length  and 
breadth  ? 

11.  There  are  two  square  fields;  a  side  of  one  is  10  rods 
longer  than  a  side  of  the  other,  and  the  areas  are  as  9 
to  4.     What  is  the  length  of  their  sides  ? 

12.  What  two  numbers  are  those  whose  product  is  135, 
and  the  difference  of  their  squares  is  to  the  square  of  their 
difference  as  4  to  i  ? 

13.  Find  two  numbers  whose  product  is  320 ;  and  the 
difference  of  their  cubes  is  to  the  cube  of  their  difference 
as  61  to  I. 


CHAPTER    XVIII. 
PROGRESSION. 

379.  A  Progression  is  a  series  of  quantities  which 
increase  or  decrease  according  to  a  fixed  law. 

380.  The  Terms  of  a  Progression  are  the  quan- 
tities which  form  the  series.  The  first  and  last  terms  are 
the  extremes  ;  the  others,  the  means, 

381.  Progressions  are  of  three  kinds:  arithmetical, 
geometrical,  and  Tiarmonicah 

ARITHMETICAL    PROGRESSION. 

382.  An  Arithmetical  Progression  is  a  series 
which  increases  or  decreases  by  a  constant  quantity  called 
the  common  difference. 

383.  In  an  ascending  series,  each  term  is  found  ly  adding 
the  common  difference  to  the  preceding  term. 

If  the  first  term  is  «,  and  the  common  difEerence  d,  the  series  is 

a,  a+d,  a  +  2d,  a  +  ^d,  etc. 
If  a  =  2,  and  d  =  2,  the  series  is  2,  5,  8,  11,  14,  etc. 

384.  In  a  descending  series,  each  term  is  found  by 
sultr acting  the  common  difference  from  the  preceding  term. 

If  a  is  the  first  term,  and  d  the  common  difference,  the  series  is 
a,  a—d,  a— 2d,  a— 3d,  etc. 

In  this  case,  the  common  difference  may  be  considered  —d.  Hence, 
the  common  difference  may  be  either  positive  or  negative.  And,  since 
adding  a  negative  quantity  is    equivalent  to    subtracting   an  equal 

379.  What  is  a  progression?  380.  The  terms?  381.  How  many  kinds  of 
progrepsion?  382.  An  arithmetical  i)rogre8Pion  ?  What  is  this  constant  quantity 
•ailed  ?    383.  An  ascending  series  ?    384.  A  descending  series  ? 


306  AEITHMETICAL     PROGKESSION. 

positive  one,  it  may  therefore  properly  be  said  that  each  successive 
term  of  the  series  is  derived  from  the  preceding  by  the  addition  of  tlie 
common  difference.    (Art.  75,  Prin.  3.) 

Notes.— I.  The  common  difference  was  formerly  called  arithmetical 
ratio;  but  this  term  is  passing  out  of  use. 

2.  An  Arithmetical  Progression  is  sometimes  called  an  Equidifferent 
Series,  or  a  Progression  by  Difference.  In  every  progression  there  may 
be  an  infinite  number  of  terms. 

385.  If  four  quantities  are  in  arithmetical  progression, 
the  sum  of  the  extremes  is  equal  to  the  sum  of  the  means. 

Let    a,  a  +  d,  a  +  2d,  a  +  3d,    be  the  series. 
Adding  extremes,  etc.,    2a  +  3d  =  2a+sd. 

Or,  let    2,  2  +  3,  2  +  6,  2  +  9,    be  the  series. 
Then      2  +  2  +  9  =  2  +  3  +  2  +  6. 

386.  If  three  quayitities  are  in  arithinetical  progression, 
the  sum  of  the  extremes  is  equal  to  double  the  mean. 

Let    a,  a  +  d,  a  +  2d,  be  the  series. 
Then    2a  +  2d  =  2{a  +  d), 

Again,  let    2,2  +  4,2  +  8,    be  the  series. 
Then    2  +  2  +  8  =  2(2  +  4). 

Cor. — An  Arithmetical  Mean  between  two  quantities 
may  be  found  by  taking  half  their  sum. 

387.  In  Arithmetical  I^rogression  there  are  five 
elements  to  be  considered:  the  first  term,  the  common 
difference,  the  last  term,  the  number  of  terms,  and  the  sum 
of  the  terms. 

Let       a  =  the  first  term. 

d  =  the  common  difierence, 
I  =  the  last  term. 
n  =  the  number  of  terms. 
8  =  the  sum  of  the  terms. 

The  relation  of  these  five  quantities  to  each  other  is  such 
that  if  any  three  of  them  are  given,  the  other  two  can  be 
found. 

385.  What  is  true  of  four  quantities  in  arithmetical  progression  ?  386.  Of  three 
quantities?  387.  Name  the  elements  in  arithmetical  progression?  What  relation 
have  they  to  each  other  { 


ARITHMETICAL     PKOGRESSION.  207 


CASE     I. 

388.  The  First  Term,  the  Common  Difference,  and  Number 
of  Terms  being  given,  to  Find  the  Last  Term. 

Each  succeeding  term  of  a  progression  is  found  by  adding  the 
common  difference  to  the  preceding  term.  (Art.  384.)  Therefore  the 
terms  of  an  ascending  series  are 

a,    a+d,    a+2d,    a+sd,    etc. 
The  terms  of  a  descending  series  are 

a,    a—d,    a — 2d,    a— 3d,    etc. 
It  will  be  seen  that  the  coefficient  of  d  in  each  term  of  both  series 
is  one  less  than  the  number  of  that  term  in  the  series.    Therefore, 
putting  I  for  the  last  or  nth.  term,  we  have 

Formula  I.       I  =  a  ±{n—  i)d. 

EuLE. — I.  Multiply  the  common  difference  hy  the  number 
of  terms  less  one. 

II.  When  the  series  is  ascending,  add  this  product  to  the 
first  term  ;  when  descending,  subtract  it  from  the  first  term, 

1.  Given  «  =  3,  d=  2,  and  n-=i,  to  find  I. 

l  =  a±  (n—i)d  =  3  +  (7—1)2  =  15,  Ans. 

2.  Given  «  =  25,  ^  =  —  2,  and  w  =  9,  to  find  I 

3.  Given  az=  12,  d  =  4,  and  n  =  15,  to  find  L 

4.  Given  a  =  1,  d=  —^,  and  n  =13,  to  find  t 

5.  Given  «  =  |,  d  =  \,  and  n  =  g,  to  find  I. 

6.  Given  «  =  i,  d=^  —  .01,  and  n  =  10,  to  find  h 

7.  Find  the  12th  term  of  the  series  3,  5,  7,  9,  11,  etc 
Note. — In  this  problem,  a  =  3,  d  —  2,  n  =  12.    Ans.  25. 

8.  Find  the  15th  term  of  i,  4,  7,  10,  etc. 

9.  Find  the  9th  term  of  31,  29,  27,  25,  etc. 

10.  What  is  the  30th  term  of  the  series  i,  2  J,  4,  5I,  etc. 

11.  Find  the  25th  term  of  the  series  x  +  ^x  +  ^x-^-yx,  etc. 

12.  Find  the  nth  term  of  the  series  2a,  sa,  Sa,  iia,  etc. 

388.  What  is  the  rule  for  finding  the  last  term  ? 


208  AKITHMETICAL     PROGEESSIOIJ", 


CASE    II. 

389.  The  Extremes  and  Number  of  Terms  being  given,  tc 
Find  the  Sum  of  the  Series. 

Let  a,  a  +  d,  a  +  2d,  a  +  3d  .  .  .  I,  be  an  arithmetical  progression, 
the  sum  of  which  is  required. 

Since  the  sum  of  two  or  more  quantities  is  the  same  in  whatever 
order  they  are  added  (Art.  63,  Prin,  2),  we  have 

8  =  a+  {a+  d)  +  {a+  2d)  +  (a+  3d)  +  ,  ,  .  +1 
Inverting,  8  =  1  +  {I— d)  +  {I  —  2d)  +  {I  —  3d)  +  .  .  .  +a 
Adding,         28  =  a  +  I  +  {a  +  I)  +  {a+l)  +  {a  +  l)  +  .  .  .  +a+l 

.'.    28  =  (a  +  T)  taken  n  times,  or  as  many  times  as  there  are 
terms  in  the  series. 
That  is,         28  =  {a  +  l)n.    Hence,  the 

Formula  II.       s  =  i — ^t_Z  x  n. 

2 

EuLE. — Multiply  half  the  sum  of  the  extremes  ly  the 
number  of  terms. 

Cor. — From  the  preceding  illustration  it  follows  that  the 
sum  of  the  extremes  is  equal  to  the  sum  of  any  two  terms 
equally  distant  from  the  extremes. 

Thus,  in  the  series,  3,  5,  7,  9.  11,  13,  the  sum  of  the  first  and  last 
terms,  of  the  second  and  fifth,  etc.,  is  the  same,  viz.,  16. 

1.  Given  «  =  4,  /  =  148,  and  n  =  15,  to  find  5. 
Solution.    4+148  =  152,  and  (152-5-2)  x  15  =  1140,  Arts, 

2.  Given  a  =  ^,  I  =  30,  and  n  =  50,  to  find  s. 

3.  Given  a  =  6,  Z  =  42,  and  n  =    9,  to  find  5. 

4.  Given  «  =  5,  ?  =  75,  and  n  =  35,  to  find  s, 

5.  Given  a  =  2,  I  =    i,  and  w=  17,  to  find  s. 

6.  Find  the  sum  of  the  series  2,  5,  8, 1 1,  etc.,  to  20  terms. 

7.  Find  the  sum  of  the  series  i,  i^,  2,  2^,  etc.,  to  25  terms. 

8.  Find  the  sum  of  the  series  75,  72,  69,  66,  63,  etc., 
to  15  terms. 


ARITHMETICAL     PRO  G  R  E  SSI  OIT.  209 

390.  The  two  preceding  formulas  are  fundamental,  and 
furnish  the  means  for  solving  all  the  problems  in  Arith- 
metical Progression.  From  them  may  be  derived  eighteen 
other  formulas. 

By  Formula  L 

391.  This  formula  contains /owrtfiyere/i^  quantities;  the 
first  term,  the  common  difference,  the  last  term,  and  the 
number  of  terms.  If  any  three  of  these  quantities  are  given, 
the  other  may  be  found.     (Art.  388.) 

I.     1  =  a  ±,  (n  —  i)  ^ /   a,cl,  and  71  being  given. 

3.  Given  d,  Z,  and  n,  to  find  a,  the  first  term.* 

Transposing  (n—i)  d  in  (i), 

a  =  l±  {n—i)d. 

4.  Given  a,  I,  and  n,  to  find  d,  the  common  difference. 

Transposing  in  (i),  and  dividing  by  {n—i), 

,      I  —  a 

a  = • 

n  — I 

5.  Given  a,  d,  and  ?,  to  find  n,  the  number  of  terms. 

Clearing  of  fractions  and  reducing  (4), 

I  —  a 

n  =  —-=—  +  I. 

d 

1.  Given  a  =  2$,  d=i   3,  and  n  =  12,  to  find  I 

2.  Given  a  =  ^S,  d  =    5,  and  n  =  45,  to  find  I 

3.  Given  d=:    3,    Z  =  35,  and  71  =    9,  to  find  a. 

4.  Given   I  =  ^y,  d  =    5,  and  71  =  21,  to  find  a, 

5.  Given  «  =  15,   1  =  85,  and  n=z  ^1,  to  find  d, 

6.  Given  a  =  28,   1=    7,  and  ?^  =  26,  to  find  ^. 

7.  Given  a=z  2^,  d  =    5,  and  Z  =  5^38,  to  find  w. 

8.  Given  a=    6,  t?=    6,  and  Z=ii52,  to  find  w. 

*  For  Formula  2' see  Art.  389. 


21(/  AEITHMETICAL     PROGRESSION. 

By  Formula  II. 

392.  In  this  formula  there  are  four  different  quantities: 
the  first  term,  the  last  term,  the  number  of  terms,  and  the 
sum  of  the  terms.  If  any  three  of  these  quantities  are 
given,  the  other  may  be  found.     (Art.  389.) 

2.    s  = X  iif    a,  I,  and^^^  being  given. 

Note. — For  Formulas  3-5,  see  Article  391. 

6.  Given  I,  n,  and  s,  to  find  a,  the  first  term. 
Clearing  (2)  of  fractions,  dividing  and  transposing, 

a  = 1. 

n 

7.  Given  a,  n,  and  s,  to  find  I,  the  last  term. 

Transposing  in  (6),  we  have 

I  = a. 

n 

8.  Given  a,  I,  and  5,  to  find  n,  the  number  of  terms. 
Clearing  (7)  of  fractions,  transposing,  factoring,  and  dividing, 

2.S 

w  =  ', 

a  ^-l 

1.  Given  «=   9,  7  =  41,  and  w=     7,  to  find  5. 

2.  Given  «  =    \,  ?  =  45,  and  n  =    50,  to  find  s. 

3.  Given  ^  =  50,  d:=   4,  and  w  =    12,  to  find  a. 

4.  Given  a=    9,  Z  =  41,  and  5  =  150,  to  find  71. 

5.  Given  d=z    7,  ?=2i,  and  ?^  =    35,  to  find  05. 

6.  Given  a  =  46,  1=  24,  and  5  =  455,  to  find  n. 

7.  Given  a  =  27,  w  =    9,  and  5  =    72,  to  find  I 

8.  Given  a  =  72,  n=    8,  and  s  =  288,  to  find  I 

9.  Find  the  sum  of  the  series  3,  5,  7,  9,  etc.,  to  15  terms. 

10.  Find  the  twentieth  term  of  5,  8,  11,  14,  17,  etc. 

11.  If  the  first  term  of  an  ascending  series  is  5,  and  the 
common  difference  4,  what  is  the  15th  term? 


ARITHMETICAL     PROGRESSION. 


211 


393.  The  remaining  twelve  fonnulas  are  derived  by 
combining  the  preceding  ones  in  such  a  manner  as  to 
eliminate  the  quantity  whose  value  is  not  sought.  They 
are  contained  in  the  following 

TABLE. 


No. 


GlVBN. 


Required. 


FOBUUXAS. 


9- 
lo. 
II. 

12. 

14. 

15- 

16. 

17. 
18. 
19. 
20. 


d  n,  s 

d,   I,  s 

a,   I,  s 

I,  n,  s 

a,  n,  8 

d,  n,  s 

a,  d,  s 

a,  df  s 

d,    I,  s 

a,  d,  n 

a,  d,  I 

d,  I,  n 


a  = 


25  —  dn^  +  dn 


271 


"^^f^yO+i/"'^' 


_  2(nl  —  s) 
w  (w—  i) 

25  —  2an 


d  = 
d=z 
d  = 


n^  —  n 


±^(2^—^)24-  Sds—2a-{-d 


71=1 


n=z 


2d 


2l+d±  V(2l  ^df—Ms 


2d 


n 


5  =  -  [2«  +  (^^  —  i)  ^ 

Ij^  a      Z2  _  ^2 

5  = Y i— 

2  2d 

5  =  -  [2?—  (7^—  i)cZ] 


._  Of  the  twenty  fonnulas  in  Arithmetical  Progression,  the  jirsA 
two  are  indispensable,  and  should  be  thoroughly  committed  to  memory ; 
the  next  six  are  important  in  the  solution  of  particular  problems.  The 
remaining  twelve  are  of  less  consequence,  but  will  be  fountJ 
interesting  to  the  inquisitive  student. 


212  ARITHMETICAL     PROGRESSION. 

394.  By  the  fourth  formula  in  x\rt.  391,  any  number  of 
arithmetical  means  may  be  inserted  between  two  given 
terms  of  an  arithmetical  progression.  For,  the  number  of 
terms  consists  of  the  two  extremes  and  aU  the  intermediate 
terms. 

Let  m  =  the  number  of  means  to  be  inserted. 

Then  m  +  2  =  n,  the  whole  number  of  terms. 

Substituting  m  +  2  f or  n  in  the  fourth  formula,  we  have 

d  = .    Hence, 

m+i 

27ie  required  number  of  means  is  found  by  the  continued 
addition  of  the  common  difference  to  the  successive  terms. 

1.  Find  4  arithmetical  means  between  i  and  31. 

2.  Find  9  arithmetical  means  between  3  and  48. 


PROBLEMS. 

1.  If  the  first  term  of  an  ascending  series  is  5,  the  common 
difference  3,  and  the  number  of  terms  15,  what  is  the  last 
term  ? 

2.  If  the  first  term  of  a  descending  series  is  27,  the 
common  difference  3,  and  the  number  of  terms  12,  what  is 
the  last  term  ? 

3.  If  the  first  term  of  an  ascending  series  be  7,  and  the 
common  difference  5,  what  will  the  20th  term  be  ? 

4.  Find  5  arithmetical  means  between  2  and  60. 

5.  What  is  the  sum  of  100  terms  of  the  series  -J-,  f,  i,  f, 
h  2,  i,  h  3»  etc. 

6.  If  the  sum  of  an  arithmetical  series  is  18750,  the  least 
term  5,  and  the  number  of  terms  20,  what  is  the  common 
difference  ? 

7.  Required  the  sum  of  bhe  odd  numbers  i,  3,  5,  7,  9,  11, 
etc.,  continued  to  76  terms? 

8.  Required  the  sum  of  100  terms  of  the  series  of  even 
numbers  2,^4,  6,  8,  10,  etc. 


ARITHMETICAL     PRO  GRESSI  0:N^.  213 

9.  The  extremes  of  a  series  are  2  and  47,  and  the  number 
of  terms  is  10.     What  is  the  common  difference  ? 

10.  Insert  8  means  between  6  and  72. 

11.  Insert  9  means  between  12  and  108. 

12.  The  first  term  of  a  descending  series  is  100,  the 
common  difference  5,  and  the  number  of  terms  15.  What 
is  the  sum  of  the  terms  ? 

Note. — i.  In  Arithmetical  Progression,  problems  often  occur  in 
which  the  terms  are  not  directly  given,  but  are  implied  in  the 
conditions.  Such  problems  may  be  solved  by  stating  the  conditions 
algebraically,  and  reducing  the  equations. 

13.  Find  four  numbers  in  arithmetical  progression,  whose 
ium  shall  be  48,  and  the  sum  of  their  squares  656. 

Let  X  =  the  second  of  the  four  numbers. 

And  1/  =  their  common  difference. 

By  the  conditions,  (a^— y)  +  x  +  {x  +  i/)  +  (x  +  2y)  =   48      (i) 

And  (ps—yf  +  x^  +  {x+ yf  +  {x+ 2y*  =  656      (2) 

Uniting  terms  in  (i),  42; +  2^=    48      (3) 

"      "(2),  4»'  +  4^+62/«  =  656      (4) 

Transposing  and  dividing  in  (3),  y  =  2^  —  2x        (5) 

Dividing  (4)  by  2,  27l^  +  2xy+2f  =  328      (6) 

Substituting  value  of  y,  2X^  +  20(24—20;)  +  3(24 — 2xY  =  328 
Reducing,  ic*— 2405  = —140 

Completing  square,  etc.,  a;  =    14  or  10 

Substituting  in  (5)  y  =  —  4  or    4 

Hence  the  required  numbers  are  6,  10,  14,  and  18. 

Note. — 2.  The  first  two  values  of  x  and  y  produce  a  descending  series ; 
the  other  two  an  ascending  series.     In  both  the  numbers  are  the  same. 

14.  Find  three  numbers  in  arithmetical  progression  whose 
Bum  is  15,  and  the  sum  of  their  cubes  is  495. 

15.  If  100  marbles  are  placed  in  a  straight  line  a  yard 
apart,  how  far  must  a  person  travel  to  bring  them  one  by 
one  to  a  box  a  yard  from  the  first  marble  ? 


214  AKITHMETICAL     PROGRESSION-. 

1 6.  How  many  strokes  does  a  common  clock  strike  in 
?4  hours? 

17.  A  student  bought  25  books,  and  gave  10  cents  for  the 
first,  30  cents  for  the  second,  50  cents  for  the  third,  etc. 
What  did  he  pay  for  the  whole  ? 

18.  A  boy  puts  into  his  bank  a  cent  the  first  day  of  the 
year,  2  cents  the  second  day,  3  cents  the  third  day,  and  so 
on  to  the  end  of  the  year.  What  sum  does  he  thus  lay  up 
in  365  days? 

19.  The  clocks  of  Venice  go  on  to  24  o'clock.  How  many 
strokes  does  one  of  them  strike  in  a  day  ? 

20.  What  will  be  the  amount  of  li,  at  6  per  cent  simple 
interest,  in  20  years  ? 

21.  What  three  numbers  are  those  whose  sum  is  120,  and 
the  sum  of  whose  squares  is  5600  ? 

22.  A  traveller  goes  10  miles  a  day ;  three  days  after, 
another  follows  him,  who  goes  4  miles  the  first  day,  5  the 
second,  6  the  third,  and  so  on.  When  will  he  overtake  the 
first? 

23.  Find  four  numbers,  such  that  the  sum  of  the  squares 
of  the  extremes  is  4500,  and  the  sum  of  the  squares  of  the 
means  is  4100. 

24.  A  sets  out  from  a  certain  place  and  goes  i  mile  the 
first  day,  3  miles  the  second  day,  5  the  third,  etc.  After  he 
has  been  gone  3  days,  he  is  followed  by  B,  who  goes  1 1  miles 
the  first  day,  12  the  second,  etc.     When  will  B  overtake  A  ? 

25.  The  first  term  of  a  decreasing  arithmetical  progression 
is  10,  the  common  difference  |,  and  the  number  of  terms  21. 
Required  the  sum  of  the  series. 

26.  A  debt  can  be  discharged  in  60  days  by  paying  $1  the 
first  day,  $4  the  second,  $7  the  third,  etc.  Required  the 
amount  of  the  debt  and  of  the  last  payment 


GEOMETEICAL     PEO  G  EES  SIGN .  215 


GEOMETRICAL    PROGRESSION. 

395.  A  Geometrical  Progj^ession  is  a  series  of 
quantities  which  increase  or  decrease  by  a  constant  multiplier 
called  the  ratio.    Hence, 

The  ratio  may  be  an  integer  or  o^  fraction. 

Note, — When  the  ratio  'ib  fractional,  the  series  will  decrease.  For 
multiplying  by  a  fraction  is  taking  a  certain  part  of  the  multiplicand 
as  many  times  as  there  are  like  parts  of  a  unit  in  the  multiplier. 

396.  In  a  geometrical  series,  each  succeeding  term  is 
found  by  multiplying  the  preceding  one  by  the  ratio. 

Thus,  if  o  is  the  first  term,  and  r  the  ratio,  the  series  is 

a,    or,    ar^    ar*,    or*,    ar^y    ar^,    etc. 
If  the  ratio  is  3,  the  series  is 

a,    ax 3,    «x3*,    ax3«,    etc. 
If  the  ratio  is  |,  the  series  is 

a,    ax^,    axjx^,    oxjx^x^^,    etc. 

397.  An  Ascending  Series  is  one  which  increases 
by  an  integral  ratio ;  as,  2,  4,  8,  16,  32,  etc. 

398.  A  Descending  Series  is  one  which  decreases 
by  a  fractional  ratio;  as,  64,  32,  16,  8,  etc. 

399.  When  the  ratio  is  a  positive  quantity,  aU  the  terms 
of  the  progression  are  positive;  when  it  is  negative,  the 
terms  are  alternately  positive  and  negative. 

Thus,  if  the  first  term  is  a,  and  the  ratio  —3,  the  series  is 
«»     — 3«»     +9«»    —2705,     +Sia,    etc. 

395.  Wliat  l8  a  geometrical  progression  ?  397.  What  Is  an  ascending  serlM  f 
J98.  Descending?    399.  Wlxat  law  governs  the  signs? 


216  GEOMETBICAL     PROGRESSION. 

400.  In  geometrical  progression  there  are  five  elements, 
the  first  term,  the  last  term,  the  number  of  terms,  the 
common  ratio,  and  the  sum  of  the  terms. 

Let        a  =  the  first  term, 
/  =  the  last  term, 
n  =  the  number  of  terms, 
r  =  the  ratio, 
8  =  the  sum  of  the  terms. 

The  relation  of  these  five  quantities  to  each  other  is  such 
that  if  any  three  of  them  are  given,  the  other  two  can  be 
found. 

CASE    I. 

401.  The  First  Term,  the  Number  of  Terms,  and  the  Ratio 

being  given,  to  Find  the  Last  Term. 

In  this  problem,  a,  w,  and  r  are  given,  to  find  I,  the  last  term. 

The  successive  terms  of  the  series  are 

a,    ar,    ar^,    ar^,    ar^^    etc,  to    ar»-^    (Art.  397.) 

By  inspection,  it  will  be  seen  that  the  ratio  r  consists  of  a  regular 
series  of  powers,  and  in  each  term  the  index  of  the  power  is  one  less 
than  the  number  of  the  terms.  Therefore,  the  last  or  nth  term  of  the 
series  is  ar''-'^.     Hence,  we  have 

Formula  I.       I  =.  ar"^^. 

Rule. — Multiply  the  first  term  ly  that  'power  of  the  ratio 
vjhose  index  is  one  less  than  the  number  of  terms. 

CoR. — Any  term  in  a  series  may  be  found  by  the  preceding 
rule  ;  for  the  series  may  be  supposed  to  stop  at  that  term. 

1.  Given  a=    ^,  n  ^  6,  and  r  =  2,  to  find  I 

2.  Given  a=    2,  w  =  8,  and  r  =  s,  to  find  I. 

3.  Given  a  =  72,  71  =  5,  and  r  =  |,  to  find  I, 

4.  Given  a=    5,  7^  =  4,  and  r  =  4,  to  find  I, 

5.  Given  a=    7,  ?i  =  5,  and  r  =  2,  to  find  I. 

6.  Given  a  =  10,  n  =  6,  and  r  =  —  5,  to  find  I 

400.  Name  the  elements  In  jceometrical  prog^Bsion.  401.  How  find  the  last 
term? 


GEOMETRICAL     P  ROGRESSIOlif  217 


CASE    II. 

402.  The   First  Term,  the  Last  Term,  and  the  Ratio  being 
given,  to  Find  the  Sum  of  the  Terms. 

In  this  problem,  a,  I,  and  r  are  given,  to  find  8, 

Since  8  =  the  sum  of  the  terms,  we  have 

8  =  a-\-a/r  +  ar'^  +  a7^-{-  ....  +a7^-2  +  ar»-\  (i) 

Multiplying  (i)  by  r, 

rs  =  ar  +  ar^  +  a7^  +  ar^+ . . . . +aT^^  +  ar*.  (2) 

Subtracting  (i)  from  (2),         ra—s  =  ar*—a.  (3) 

Factoring  and  dividing,  a  = •  (4) 

In  equation  (4),  ar^  is  the  last  term  of  (2),  and  is  therefore  the 
product  of  the  ratio  by  the  last  term  in  the  given  series. 
Substituting  Ir  for  ar^,  we  have 

Ir  -~a 


Formula  II. 


r—  I 


EuLE. — Multiply  the  last  term  hy  the  ratio,  from  the 
product  subtract  the  first  term,  and  divide  the  remainder  by 
the  ratio  less  one. 

PW'  For  the  method  of  finding  the  sum  of  an  infinite  descending 
aeries,  see  Art.  435. 

1.  Given  a  =  2,  ?  =  500,  and  r  =  3,  to  find  the  sum. 

_                          Ir—a      500  X  3  —  2  . 

Solution.     «  = = =  749,  Ans, 

2.  Griven  «  =  3,  1=^  9375?  and  r  =  5,  to  find  s, 

3.  Given  a  =  9,  l^  9000,  and  r  =  10,  to  find  s. 

4.  Given  a  =  s^  ^  =  20480,  and  r  ^  4,  to  find  s. 

5.  Given  a  =:  15,  1=  3240,  and  r  =  6,  to  find  s. 

6.  Given  a  =  2^,  1=  6400,  and  r  =z  4,  to  find  5. 

402.  How  find  the  sum  of  the  terms  ? 
10 


218  GEOMETRICAL     PE0GRESSI02T. 

403.  The  two  preceding  formulas  furnish  the  means  loi 
solving  all  problems  in  geometrical  progression.  Thej  may 
be  varied  so  as  to  form  eighteen  other  formulas. 


By  Formula  I. 

404.  The  first  formula  contains /owr  different  quantities: 
the  first  term,  the  last  term,  the  ratiOy  and  the  number  of 
terms.  If  any  three  of  these  quantities  are  given,  the  oiuer 
may  be  found.    By  the  first  formula, 

I.    I  =  ar**^^;    a,  n,  and  r  being  given.    (Art.  401.) 
For  formula  2,  see  Article  402. 

3.  Given  Z,  n,  and  r,  to  find  or,  the  first  term. 

Factoring  (i),  and  dividing  by  y-^ 
/ 

4.  Given  a,  ?,  and  7^,  to  find  r,  the  ratio. 

Dividing  (i)  by  «,  and  extracting  the  root  denoted  by  the  index, 

5*  Given  a,  ?,  and  r,  to  find  n,  the  number  of  terms. 

I 


a 


Dividing  (i)  by  a,  r""^ 

By  logarithms,         log  r  (n—  i)  =  log  ?  —  log  a 

-.      ,  log  I  —  log  a 

Dividing,  etc.,  n  =  —-i^-^—  +  ^' 

Note.— Sin«e  this  formula  contains  logarithms,  it  may  be  deferred 
till  that  subject  is  explained. 

1.  Given  «  =  3,  n  =  $,  and  r  =  10,  to  find  I 

2.  Given  a  =  5,  n  =  6y  and  r  =  5,  to  find  /.  - 

3.  Given   Z  =1  256,  n  =  S,  and  r  =  2,  to  find  a. 

4.  Given   I  =  243,  ^  =  5,  and  r  =  3,  to  find  a. 

5.  Given  a  =  2^  Z  =  2592,  and  /i  =  5,  to  find  n 

6.  Given  a  =  4,  /  :=  2500,  and  w  =  5,  to  find  n 


GBOMETKICAL     PRO  GRESSIOl?'.  219 


By  Formula  IL 

405.  This  formula  contains  four  different  quantities :  the 
fird  term,  the  last  term,  the  ratio,  and  the  sum  of  ths 
terms.  If  a7iy  three  of  them  are  given,  the  other  may  be 
found.    By  the  second  formula, 

1/p ft 

2,    s  =  — ,    a,  I,  and  /•  being  given.     (Art.  402.) 

For  formulas  3-5,  see  Article  404. 

6.  Given  I,  r,  and  s,  to  find  a,  the  first  term. 

Clearing  (2)  of  fractions,  etc., 

a  =  Ir  —  s  {r  —  i) 

7.  Given  a,  r,  and  5,  to  find  Z,  the  last  term. 

Transposing  in  (6), 

;r  =  a  +  s  (r  —  i). 
Dividing  by  r, 

/  —  «  +  g(r~»i) 

8.  Given  a,  7,  and  s,  to  find  r,  the  ratio. 

Clearing  (2)  of  fractions, 

sr  —  s-=  Ir  —  a. 
Transposing  in  the  last  equation, 

sr  —  Ir  =  s  —  a. 

Factoring,  etc., 

8  — a 
r  = 


8-1 


1.  Given  a  =^  2,  I  z=  162,  and  r  =  3,  to  find  s, 

2.  Given    /  =1  54,  r  =  3,  and  s  =z  80,  to  find  a. 

3.  Given  o^  =  4,  r  =  5,  and  s  z^  624,  to  find  I. 

4.  Given  a  =  4,  I  =  12500,  and  s  =  15624,  to  find  n 

5.  Given  a  =  5,  I  =  180,  and  r  ^  6,  to  find  s. 

6.  Given  a  =  t,  r  =  ^,  and  s  =  847,  to  find  I, 


220 


GEOMETRICAL     PROGRESSION?-, 


406.  The  remaining  twelve  formulas  are  derived  by 
combining  the  preceding  ones  in  such  a  manner  as  to 
eliminate  the  quantity  whose  value  is  not  sought. 


TABLE. 


No 


Given. 


Requered. 


FOKMULAS. 


9 

lO 

II 

12 

13 

14 

15 

16 

17 

18 
19 
20 


n,  r,  s 

I,  n,  s 

a,  n,  s 

n,  r,  s 

a,   I,  s 

a,  r,  s 

h  r,  s 

tty  71,  S 

I,  71,  S 

a,  n,  r 

I,  n,  r 

a,  I,  Tfi 


a  = 


{r—\)s 


/•"  —  I 

a{s  —  ay-^  =  /  (5  —  iy-\ 

l{s  —  ly-^  =  a{s  —  ay-K 

r"  —  I 

71  =  log  ^- log  Q^  ^ 

log(s—a)—log(s—iy 
]og[a  +  {r—i)s]  —  log  a 
log  r 

_log|— log  [Ir—  (r— 1)5] 

~"  log  r 


n 


+  1 


s  s 

r" r  =  1 

a  a 


/•"  + 


%-i 


I—  S  1  —  8 

a(r''  —  i) 


r  - 

- 1 

^/i  — "  v« 


Of  the  twenty  formulas  in  Geometrical  Progression,  the  fird 
two  are  fundamental,  and  should  be  thoroughly  committed  to  memory ; 
the  next  six  are  important  in  the  solution  of  particular  prohlems.  The 
remainder  are  less  practical. 


GEOMETRICAL     PK  OGRE  SSI  ON.  221 

407.  By  the  fourth,  formula  (Art.  404),  any  number  of 
geometrical  means  may  be  found  between  two  given 
quantities. 

Let  m  =  the  number  of  means  required. 

Then  m  +  2  =  n. 

Substituting  m  +  2  for  n  in  the  formula,  we  have 


'•  =  ©-^ 


The  ratio  being  found,  the  means  required  are  obtained  by  continued 
multiplication. 

1.  Find  two  geometrical  means  between  3  and  192. 

Solution,    r  =  Q*  =  j/i^?  =  ^ej  =  4. 

The  ratio  being  4,  the  first  mean  is  3  x  4  =  12  ;  the  second  is 
12x4  =  48. 

2.  Find  three  geometrical  means  between  |  and  128. 


PROBLEMS. 

1.  In  a  geometrical  progression,  the  first  term  is  6,  the 
last  term  2916,  and  the  ratio  3.  What  is  the  sum  of  all  the 
terms  ? 

2.  In  a  decreasing  geometrical  series,  the  first  term  is  |, 
the  ratio  i,  and  the  number  of  terms  8.  What  is  the  sum 
of  the  series  ? 

3.  What  is  the  sum  of  the  series  i,  3,  9,  27,  etc.,  to  15 
terms? 

4.  Find  the  sum  of  12  terms  of  the  series,  i,  |,  f,  -jy,  etc. 

5.  If  the  first  term  of  a  series  is  2,  the  ratio  3  and  the 
number  of  terms  15,  what  is  the  last  tenn  ? 

6.  What  is  the  i6th  term  of  a  series,  the  first  term  of 
which  is  3,  and  the  ratio  3  ? 

Note. — When  the  terms  of  the  series  are  not  stated  directly,  they 
may  be  represented  algebraically. 


222  GEOMETEICAL     PKOGRESSIOK. 

7.  Find  three  numbers  in  geometrical  progression,  such 
that  their  sum  shall  be  2 1,  and  the  sum  of  their  squares  189. 

Let  the  three  numbers  be  x,  ^/xy,  and  y. 

By  the  conditions,           a;+  ^/m-\-y  =  21  (i) 

And                                    a;2  +  a;y+y2  =  189  (2) 

Transposing  and  sq.  (i),  as^  +  20;^ + ^^2  =  441  _  42  /y/^ + xy  (3) 

Subtracting  (2)  from  (3),                 iry  =  25 2  - 42  'v/^+  xy  (4) 

Transposing,  etc.,                        ^/xy  =  6  (5) 

Involving,                                  ^       icg^  =  36  (6) 

And                                                 32^  =  108  (7) 
Subtracting  (7)  from  (2),  c^—2xy+y^  =    81 

Extracting  root,                           x—y  =     9  (8) 

Substituting  (5)  in  (i),                 x+y  =    15  (9) 
Combining  (8)  and  (9),                      a?  =    12 

y=     3 
Hence  the  numbers,  12,  6,  and  3,  Ans. 

8.  A  father  gives  his  daughter  $1  on  New  Year's  day 
t(>vrards  her  portion,  and  doubles  it  On  the  first  day  of  every 
month  through  the  year.     What  is  her  portion  ? 

9.  A  dairyman  bought  10  cows,  on  the  condition  that  he 
should  pay  i  cent  for  the  first,  3  for  the  second,  9  for  the 
third,  and  so  on  to  the  last.  What  did  he  pay  for  the  last 
cow  and  for  the  ten  cows? 

10.  A  man  buys  an  umbrella,  giving  i  cent  for  the  first 
brace,  2  cents  for  the  second  brace,  4  for  the  third,  and  so 
on,  there  being  10  braces.  What  is  the  cost  of  the 
umbrella  ? 

11.  The  sum  of  three  numbers  in  geometrical  progression 
is  26,  and  the  sum  of  their  squares  364.     Find  the  numbers. 

12.  What  would  be  the  price  of  a  horse,  if  he  were  to  be 
sold  for  the  32  nails  in  his  shoes,  paying  i  mill  for  the  first, 
2  mills  for  the  second,  4  for  the  third,  and  so  on  ? 

13.  Find  four  numbers  in  geometrical  progression,  such 
that  the  sum  of  tlie  first  three  is  130,  and  that  of  the  last 
three  is  390. 


GEOMETRICAL     PROGRESSION^.  223 

14.  A  man  divides  $210  in  geometrical  progression  among 
three  persons;  the  first  had  $90  more  than  the  last.  How 
much  did  each  receive  ? 

15.  There  are  five  numbers  in  geometrical  progression. 
The  sum  of  the  first  four  is  468,  and  that  of  the  last  four  is 
2340.     What  are  the  numbers? 

16.  The  sum  of  $700  is  divided  among  4  persons,  whose 
shares  are  in  geometrical  progression ;  and  the  difference 
between  the  extremes  is  to  the  difference  between  the  means 
as  37  to  12.     What  are  the  respective  shares? 

17.  The  population  of  a  town  increases  annually  in 
geometrical  progression,  rising  in  four  years  from  loooo  to 
1 464 1.     What  is  the  ratio  of  annual  increase  ? 

18.  The  sum  of  four  numbers  in  geometrical  progression 
Is  15,  and  the  sum  of  their  squares  85.  What  are  the 
numbers  ? 


HARMONICAL    PROGRESSION.* 

408.  An  Harmonical  I^rogression  is  such,  that 
of  any  three  consecutive  terms,  the  first  is  to  the  third  as  the 
difference  of  the  first  and  second  is  to  the  difference  of  the 
second  and  third. 

Thus,  10,    12,    15,    20,    30,    60, 

are  in  harmonic  progression  ;  for 

ID  ;  15  ::  12—10  :  15—12 
12  :  20  ::  15—12  :  20—15 
15  :  30  : :  20—15  '•  30—20 
20  :  60  : :  30—20  :  60—30 
Let  a,  6,  c,  d,  e,  /,  g,  be  an  harmonical  progression,  then 

a  :  c  :  a—b  :  b—c,  etc. 
Note.— When  three  quantities  are  such,  that  the  first  is  to  the 
tJiird  as  the  difference  of  the  first  and  second  is  to  the  difference  of  the 
second  and  tliird,  they  are  said  to  be  in  Harmonical  Proportion. 
Thus,  i,  3,  and  6,  are  in  harmonical  proportion, 

408.  What  is  an  harmonical  progression  ? 
"•'  If  a  musical  string  be  divide.]   in  harmonical  proportion,  the 
/ifferent  parts  will  vibrate  in  harmony.     Hence,  the  name. 


224  HARMONICAL     PROGRESSIOK. 

409.    To    Find    the    Third    Term    of   an    Harmonical 
Progression,  the  First  Two  being  given. 

Let  a  and  h  be  the  first  two  terms,  and  x  the  third  term. 

Then  a  :  x  ::  a-l  :  l-x 

Multiplying  extremes,  etc.,  ab—ax  =  ax—hx 

Transposing,  etc.,  2ax—bx  =  db 

Factoring,  and  dividing  by  2a— h,  we  have  the 

ab 


FORMtJLA.  X 


2a  —  h 


Rule. — Divide  the  product  of  the  first  two  tenns  hy  twice 
the  first  mi?ms  the  second  term;  the  quotient  will  le  tht 
third  term. 

Note. — This  rule  furnishes  the  means  for  extending  an  harmonic 
progression,  by  adding  one  term  at  a  time  to  the  two  preceding  terms. 

1.  Find  the  third  term  in  the  harmonic  series  of  which 
12  and  8  are  the  first  two  terms.  Ans.  6. 

2.  Find  the  third  term  in  the  harmonic  series  of  which 
12  and  1 8  are  the  first  two  terms.  Ans.  2,^. 

3.  If  the  first  two  terms  of  an  harmonic  progression  are 
15  and  20,  what  is  the  third  term?  Ans.  30. 

4.  Continue  the  series  12,  15,  20,  for  two  terms. 

Ans.  30  and  60. 

5.  Continue  the  series  7^^,  9,  12,  for  two  terms. 

A71S.  18  and  2,^, 

410.  To  Find  a  Mean  or  IVIiddle  Term  between  Two  Terms 
of  an  Harmonic  Progression. 

Let  a  and  c  be  the  first  and  third  of  three  consecutive  terms  of  an 
harmonic  progression,  and  m  the  mean. 

Then  a  i  c  ::  a—m  .  m—e 

Mult,  extremes  and  means,        am—ac  =  ac—cm 
Transposing  and  uniting,  am  +  cm  =  2ac 

Factoring  and  dividing  by  a  +  c,  we  have  the 

„  2ac 

Formula.       m  = • 

a  +  c 


409.  How  find  the  third  term  of  an  harmonical  progression,  the  first  two  being 
tfiven? 


HARMOtsTICAL     PROGRESSION".  225 

Rule. — Divide  twice  the  product  of  the  first  and  third 
terms  hy  their  sum;  the  quotient  will  he  the  mean  or  middle 
term. 

6.  The  first  and  third  of  three  consecutive  terms  of  an 
harmonic  progression  are  9  and  1 8.  Required  the  mean  or 
middle  terra. 

Solution.       2x9x18  =  324,     and     9  + 18  =  27, 
Now  324-J-27  =  12,    Ans. 

7.  rind  an  harmonic  mean  between  12  and  20.    Ans,  15. 

8.  Eind  an  harmonic  mean  between  15  and  30.    Ans.  20. 

411.  The  Reciprocals  of  the  terms  of  an  harmonic 
progression  form  an  arithmetical  progression. 

Thus,  the  reciprocals  of  10,  12, 15,  20,  etc.,  viz., 

tV»    tV»    tV»    "eV*    ■^'     ^^^•» 
are  an  arithmetical  progression,  whose  common  difference  is  ^V 

Again,  let  a,  b,  c  be  in  harmonic  progression. 

Then  a  :  c  ::  a—b  :  b—c 

Mult,  extremes  and  means,         ah—ac  =  ac—bc 

Dividing  hj  abc,  ^  ~  ^ (^^-  3^4') 

Conversely,  the  reciprocals  of  an  arithmetical  progression 
form  an  harmonic  progression.     Thus, 

The  reciprocals  of  the  arithmetical  progression  i,  2,  3,  4,  5,  etc., 
viz.,  \,  ^,  \i\,  i,  etc.,  are  in  harmonic  progression. 

412.  If  the  lengths  of  six  musical  strings  of  equal  weight 
and  tension,  are  in  the  ratio  of  the  numbers 

i>    h     h     h    h    h    etc., 

the  second  will  sound  an  octave  above  the  first ;  the  third 
will  sound  the  twelfth  ;  the  fourth  the  double  octave,  etc. 


410.   Kow  find   a  mean  between  two  terms   of  an   harmonic  progression? 
411.  Whai  do  the  reciprocals  of  an  harmonical  prog;  ession  fonL . 


INFIl^ITE     SERIES, 


INFINITE    SERIES. 

413.  An  Infinite  Series  is  one  in  which  the  successive 
terms  are  formed  by  some  regular  law,  and  the  numher  of 
terms  is  unlimited. 

414.  A  Converging  Series  is  one  the  sura  of  whose 
terms,  however  great  the  number,  cannot  numerically 
exceed  definite  quantity. 

415.  A  Diverging  Series  is  one  the  sum  of  whose 
terms  is  numerically  greater  than  2ii\y  finite  quantity. 

416.  To  Expand  a  Fraction  into  an  Infinite  Series, 

Remark. — Any  common  fraction  whose  exact  value  cannot  be 
expressed  by  decimals,  may  be  expanded  into  an  infinite  series. 

1.  Expand  the  fraction  J  into  an  infinite  series. 

Solution.    1-5-3  =  .333333,  and  so  on,  to  infinity. 

Or,        I  ^3  =  ^3^  +  ^  +  ^^ + ^^,  etc.    Hence,  the 

EuLE. — Divide  the  numerator  hy  the  denominator* 

2.  Reduce  to  an  infinite  series. 

1  —X 

I— aj)i        (i  +  aj+a;2  +  a^+a^,  etc.,  the  quotient.    (Art.  1 70. ) 
+x 


+ic*,    etc. 
Therefore,  =  i+x+x'^  +  q?  +  v^-{-q?,  etc.,  to  infinity. 

413.  What  is  an  infinite  series?    414.  A  converging  series?    415.  Diverging? 
416.  How  expand  a  fraction  into  an  infinite  fcrie?? 


INFINITE     SERIES.  227 

Let  x  =  ^;  then  will   = =  2  ;  and  tlie  series  will  be 

^  '  i  —  xi—i 

I  +  i+i  +  i  +  iV  +  aV*  ®tc.,  tlie  sum  of  which  =  2. 

If  a?  =  4,  then  will  = =  |,  and  the  series  will  become 

**  I  —  a;      I  —  i      ^ 

i+i  +  i+^T+^T  +  ^5.  etc.  =  |. 

Notes. — i.  If  a?  is  less  than  i,  the  series  will  be  convergent . 

For,  when  x  is  Icm  than  i,  the  remainder  must  continually  decrease  ; 
therefore,  the  further  the  division  is  carried,  the  less  will  be  the 
quantity  to  be  added  to  the  last  term  of  the  quotient  in  order  to 
express  the  exact  value  of  the  fraction. 

2.  If  X  is  greater  than  i,  the  series  will  be  divergent. 

For,  when  x  is  greater  than  i,  the  remainder  must  constantly 
increase  ;  therefore,  the  farther  the  division  is  csLrried,  the  greater  will 
be  the  quantity  either  positive  or  negative  to  be  added  to  the  quotient. 

3.  Eeduce  the  fraction  to  an  infinite  series. 

Solution.    i-7-(i+a;)  =  i—x+^—a^+x*—x^  +  ,  etc. 


This  series  is  the  same  as  that  in  Ex.  2,  except  the  odd  powers 
of  X  are  negatice. 

Let  a?  =  2- ;    then  will =  f ;  which  is  equal  to  the  series 

I -i  + 1  -i  +  iV- aV  + ,  etc. 

4.  Reduce  the  fraction  to  an  infinite  series. 

I  —X 

Ans.  I  +  20;  +  2x^  -f  2x^  -f-  2X^,  etc. 

417.  A  fraction  whose  denominator  has  more  than  two 
terms,  may  also  be  expanded  into  an  infinite  series. 

5.  Expand 5  into  an  infinite  series. 

^  I  —  X  -\-  x^ 

I— a?+«^)  I  {i+x—Q^—x^+x^,  etc.,  Ans, 

x—x^ 
x—x^  +  a? 

—a?,  etc 


228  INFINITE     SERIES. 

418.  To  Expand  a  Compound  Surd  into  an  Infinite  Series, 


6.  Keduce  v  i  +  x  to  an  infinite  series. 


OPERATION. 


2+-|    +x 

a? 

+  «  +  — 
4 

X^  I  25"^ 

4         8    "^64 

2  +  a? +  -Z  +"5 —  z~»  ®tc.    Hence,  the 

4       lo  I  o       04 

EuLE.  —  Extract  the  square  root  of  the  given  sutfl 
(Art.  298.) 

7.  Expand  ^/^^2     Ans.x-t^^^^,  etc. 

8.  Expand  \/2,  or  Vi  +  i.    ^ws.  i  +  -f  —  ^  +  -jV?  ^^c* 

419.  The  Binomial  Theorem  applied  to  the  Formation  of 
Infinite  Series. 

The  Binomial  Theorem  may  often  be  employed  with 
advantage,  in  finding  the  roots  of  binomials.  For  a  root  is 
expressed  like  a  power,  except  the  exponent  of  one  is  an 
integer,  and  that  of  the  other  is  q>  fraction. 

9.  Expand  {x  4-  yY  into  an  infinite  series. 

Solution. — The  terms  without  coefficients  are 

«*»    aj~5y,    a;~«y%    a;~»^,    a;~»2^,    etc. 

1  X  —4 
The  coefficient  of  the  second  term  is  +  ^  ;  of  the  3d, =  —  J ; 

—  1  X  —  J 

of  the  4th  term,  — =  +  ^\,  etc. 

The  series  is    «*  +  ^x'^y  —  lx~^y^  +  ^^x~^y^»  etc. 
418.  How  expand  a  surd  into  an  infinite  eeriei  f 


INFINITE     SERIES.  229 

420.  When  the  index  of  the  required  power  of  a  binomial 
is  a  posiiivG  integer,  the  series  will  termiriate.  For,  the 
index  of  the  leading  quantity  continually  decreases  by  i ; 
and  soon  becomes  o;  then  the  series  must  stop.     (Art,  269.) 

421.  When  the  index  of  the  required  power  is  negaiive, 
the  series  will  never  terminate.  For,  by  the  successive 
subtractions  of  a  unit  from  the  index,  it  will  never  become 
o ;  and  the  series  may  be  continued  indefinitely. 

10.  Expand  {(^+1/)^  into  an  infinite  series,  keeping  the 
factors  of  the  coefiicients  distinct. 

^        ^  2X      2.^      2.4.6aj*      2.4.6.8a;' 

It.  Expand  VJ,  or  (i  +  i)^,  keeping  the  factors  of  th€> 
coefficients  distinct. 

Ans.  i+i— i-+_3 3.-J  3- 5- 7  t,. 

2         2.4       2.4.6        2.4.6.8        2.4.6.8.10 

422.  An  Infinite  Series  must  not  be  confounded  with  an 

Ivfinite  Quantity, 

423.  An  Infinite  Quantity  is  a  quantity  so  great 
that  nothing  can  be  added  to  it. 

424.  An  Infinite  Series  is  a  series  m  which  the 
number  of  terms  is  unlimited, 

425.  The  magnitude  of  the  former  admits  of  no  increase; 
while  in  the  latter  the  number  of  terms  admits  of  no 
increase,  and  yet  the  sum  of  all  the  terms  may  be  a  small 
quantity. 

Thus,  if  the  series  i  +  l +i  +  TV+"s'*>  ®*^'»  ^^  which  each  succeeding 
term  is  half  the  preceding,  is  continued  to  infinity,  the  sum  of  all  the 
terms  cannot  exceed  a  unit. 

426.  When  one  quantity  continually  approximates 
another  without  reaching  it,  the  latter  is  called  the  Litnit 
of  the  former. 


330  INFIiflTE     SERIES. 

427.  An  Infinitesimal  is  a  quantity  whose  value  is 
less  than  any  assignable  quantity/. 

428.  The  Sign  of  Infinity^  or  of  an  infinite 
quantity,  is  a  character  resembling  an  horizontal  figure 
eight  ( CO  ). 

The  Sign  of  an  Infinitesimal  is  zero  ( o  ). 

429.  One  infinite  series  may  be  greater  or  less  than 
another. 

Thus,  the  series  i  +  i  +  i  +  ^  +  A*  etc.,  whose  limit  is  2,  is  greater 
than  the  series  i  +  i  +  g  +  t^  +  sV*  6*^.,  whose  limit  is  i. 

430.  Since  an  infinitesimal  is  less  than  any  assignable 
quantity,  and  in  its  limit  approaches  zero,  when  connected 
with  finite  quantities  by  the  sign  -f  or  — ,  it  is  of  so  little 
value  that  it  may  be  rejected  without  any  appreciable  error. 

431.  An  infinite  series  may  be  multiplied  by  a  finite 
quantity. 

Thus,  if  the  series  222222,  etc.,  is  multiplied  by  3, 

the  product      666666,  etc.,  is  three  times  the  multiplicand. 

432.  An  infinite  series  may  also  be  divided  by  a  finite 
quantity. 

Thus,  if  the  series  888888,  etc.,  is  divided  by  2, 
the  quotient     444444,  etc.,  is  hali  the  dividend. 

433.  If  a.  finite  quantity  is  multiplied  by  an  infinitesimal, 
the  product  will  be  an  infinitesimal.  For,  with  a  given 
multiplicand,  the  less  the  multiplier,  the  less  will  be  tlie 
product.     Thus,   xxo  =z  o. 

434.  If  a  fi7iite  quantity  is  divided  by  an  infinitesimal^ 
the  quotient  will  be  infinite.     Thus,   ic  -^  o  =  00 . 

If  a  finite  quantity  is  divided  by  an  infinite  quantity,  the 
quotient  will  be  an  infinitesimal.     Thus,    a:  -^  00  =  o. 

If  an  infinitesimal  is  divided  by  a  finite  quantity,  the 
quotient  is  an  infinitesimal.     Thus,   o  -j-  2;  =  o. 

Note.— In  higher  mathematics,  the  expression  o -f- o  admits  of 
various  interpretations. 


IKFIl^ITE     SEEIES.  231 

435.  To  Find  the  Sum  of  a  Converging  Infinite  Series,  the 
First  Term  and  Ratio  being  given. 

By  the  second  formula  in  geometrical  progression,  we  liave  for  an 

increasing  series  (Art.  402), 

Ir  —  a  ar^  —  a 

8  — ,     or     • 

r—i  r— I 

In  a  decreasing  series,  tlie  ratio  r  is  less  tlian  i ;  therefore.  I  or  a/"»-i 
is  less  than  a.    (Art.  398.) 

Tliat  both  terms  of  the  fraction ,  or may  be  positive, 

T — I  r — I 

we  change  the  signs  of  both  (Art.  166),  and 

a  —  Ir 

8  = • 

i  —  r 

But,  in  a  decreasing  infinite  series,  I  becomes  an  infinitesimal,  or  o; 
therefore,  Ir  =  o.    (Art.  427.)    Hence,  rejecting  the  iuHnitesimal  from 

f  = ,  we  have  the 

Formula.       s  = • 

I  —  1* 

EULE. — Divide  the  first  term  ly  i  minus  tlie  ratio, 

1.  Find  the  sum  of  the  infinite  series 

I  +  i  +  i  +  ^7  +  A^  etc. 

2.  Required  the  sum  of  the  infinite  series 

I  — }  +  i  — i+,  etc. 

3.  Find  the  sum  of  the  series  \  +  J  +  J  +?  etc, 

4.  Find  the  sum  of  the  infinite  series  i  +  I  +  I  +*  eta 

5.  Find  the  sum  of  the  series  f  +  f  +  27  +>  etc. 

6.  Find  the  sum  of  the  series  3  +  2  +  |  -f ,  etc. 

7.  Find  the  sum  of  the  series  4  +  ¥  +  f|  +>  etc. 

8.  Find  the  sum  of  the  series  .ZZZZ^  etc. 

9.  Find  the  sum  of  the  series  .ddddd,  etc. 

10.  Find  the  sum  of  the  series  -  H -^■\ 5  +  etc. 

11.  Suppose  a  ball  to  be  put  in  motion  by  a  force  which 
impels  it  10  rods  the  first  second,  8  rods  the  next,  and  so  on, 
decreasing  by  a  ratio  of  ^  each  second  to  infinity.  Through 
what  space  would  it  move  ? 


OHAPTEE    XIX. 

LOGARITHMS* 

436.  The  LogarithTn  of  a  number  is  the  exponent  ol 
the  power  to  which  a  given  fixed  number  must  be  raised  to 
produce  that  number. 

437.  This  Fixed  Numler  is  called  the  ^ase  of  the 

system. 

Tlius,  if  3  is  the  base,  then  2  is  the  logarithm  of  9,  because  3^  =  9 ; 
and  3  is  the  logarithm  of  27,  because  3^  =  27,  and  so  on. 

Again,  if  4  is  the  base,  then  2  is  the  logarithm  of  16,  because 
4^^  =  16  ;  and  3  is  the  logarithm  of  64,  because  4^  =  64,  and  so  on. 

438.  In  forming  a  system  of  logarithms,  any  number, 
except  I,  may  be  taken  as  the  base,  and  when  the  base  is 
selected,  all  other  numbers  are  considered  as  some  power  or 
root  of  this  base.  Hence,  there  may  be  an  unlimited 
number  of  systems. 

Note. — Since  all  powers  and  roots  of  i  are  i,  it  is  obvious  that  other 
numbers  cannot  be  represented  by  its  powers  or  roots.    (Art.  289.) 

439.  There  are  tv/o  systems  of  logarithms  in  use,  the 
Napierian  system,!  the  base  of  which  is  2.718281828,  and 
the  Common  System,  whose  base  is  10.  J 

The  abbreviation  log  stands  for  the  term  logarithm. 


436.  Wliat  are  logarithms  ?  437.  What  is  this  fixed  number  called  ?  439.  Name 
the  systems  in  use.    The  base  of  each. 

*  The  term  logarithm  is  derived  from  two  Greek  words,  meaning 
the  relation  of  nwrnhers. 

f  So  called  fi-om  Baron  Napier,  of  Scotland,  who  invented  log- 
arithms in  1614. 

X  The  common  system  was  invented  by  Henry  Briggs,  an  English 
mathematician,  in  1624. 


LOGARITHMS.  233 

440.  The  Sase  of  common  logarithms  being  lo,  all 
other  numbers  are  considered  as  powers  or  roots  of  lo. 

Thus,         the  log.  of      i  is  o ;  for  lo"  equals      i  (Art.  259) ; 
**  *'        10  is  I  ;  for  10^      "         10 ; 

**  *•      icx)  is  2 ;  for  lo^      "       100 ; 

**         "    1000  is  3 ;  for  lo'      "     1000,  etc    Hence, 

The  logarithm  of  any  number  between  i  and  10  is  a 
fraction;  for  any  number  between  10  and  100,  the  logarithm 
is  I  plus  a  fraction  ;  and  for  any  number  between  100  and 
1000,  the  logarithm  is  2  plus  a  fraction,  and  so  on. 

441.  By  means  of  negative  exponents,  this  principle  may 
be  applied  to  fractions. 

Thus  (Art.  256),  the  log.  of  .1  is  —i  ;  for  10-^  equals  .1 ; 
"  '*  ,01  is —2 ;  for  10-2  "  .01; 
*'         "    .001  is —3;  for  10-^       **     .001. 

Therefore,  the  logarithms  for  all  numbers  between  i  and 
0.1  lie  between  o  and  — i,  and  are  respectively  equal  to  — i 
plus  a  fraction;  for  any  number  between  o.i  and  0.0 1,  the 
logarithm  is  —2  plus  a  fraction  ;  and  for  any  number 
between  o.oi  and  o.ooi,  the  logarithm  is  — 3  plus  a  fraction, 
and  so  on. 

Hence,  the  logarithms  of  all  numbers  greater  than  10  or 
less  than  i,  and  not  exact  powers  of  10,  are  composed  of 
two  parts,  an  mteger  and  o.  fraction. 

Thus,  the  logarithm  of    28    is    1.44716; 

and  of  .28    is    1.44716. 

442.  The  integral  part  of  a  logarithm  is  called  the 
Characteristic ;  the  decimal  ^^rt,  the  llantissa, 

443.  The  Characteristic  of  the  logarithm  of  a  whole 
number  is  one  less  than  the  number  of  integral  figures  in 
the  given  number. 

Thus,  the  characteristic  of  the  logarithm  of  49  is  i ;  that  of  495  is 
2 ;  that  of  4956  is  3 ;  that  of  6256.414  is  also  3,  etc. 

440.  What  is  the  logarithm  of  any  number  from  i  to  10?  From  10  to  100?  From 
100  to  1000?  442.  What  is  Ihe  integral  part  of  a  logarithm  called?  The  decirna.] 
part  ?    443.  What  is  the  characteristic  01  the  logarithm  of  a  whole  numlier  ? 


234  LOGARITHMS. 

444.  The  Characteristic  of  the  logarithm  of  a 
decimal  is  negative,  and  is  one  greater  than  the  number  of 
ciphers  before  the  first  significant  figure  of  the  fraction. 

Thus,  the  characteristic  of  the  logarithm  of  ^^  or  .i  is  —  i ;  that  of 
3^^o'  or  .01,  is  —  2 ;  that  of  ^i^^,  or  .001,  is  —3,  etc.    (Art.  256.) 

The  logarithm  of  .2  is  —  i  with  a  decimal  added  to  it ;  that  of  05 
!s  —  2  with  a  decimal  added  to  it,  etc. 

Note. — It  should  be  observed  that  the  characteristic  only  is  negative, 
while  the  mantissa,  or  decimal  part,  is  always  podtive.  To  indicate 
this,  the  sign  —  is  placed  over  the  characteristic,  instead  of  before  it. 

Thus,  the  logarithm  of      .2    is     i. 30103, 

"         "  *'     .05    is    2.69897,  eta 

445.  The  Decimal  Part  of  the  logarithm  of  any 
number  is  the  same  as  the  logarithm  of  the  number 
multiplied  or  divided  by  10,  100,  1000,  etc. 

Thus,  the  logarithm  of  1876  is  3.27325  ;  of  18760  is  4.27325,  etc. 


TABLES    OF    LOGARITHMS. 

446.  A  Table  of  Loffaritlmis  is  one  which  contains 
the  logarithms  of  all  numbers  between  given  limits. 

447.  The  Table  found  on  the  following  pages  gives  the 
mantissas  of  common  logarithms  to  five  decimal  places  for 
all  numbers  from  i  to  1000,  inclusive. 

The  characteristics  are  omitted,  and  must  be  supplied  by 
inspection.     (Arts.  443,  444.) 

Notes. — i.  The  first  decimal  figure  in  column  0  is  often  the  same 
for  several  successive  numbers,  but  is  printed  only  once,  and  is 
understood  to  belong  to  each  of  the  blank  places  below  it. 

2.  The  character  ( ♦  )  shows  that  the  figure  belonging  to  the  place 
it  occupies  has  changed  from  9  to  o,  and  through  the  rest  of  this  line 
the  first  figure  of  the  mantissa  stands  in  the  next  line  below. 

444.  What  Is  the  characteristic  of  the  loj]jarithm  of  a  decimal  ?  445.  What  is  the 
effect  upon  the  decimal  part  of  the  lo<?.  of  a  number,  if  the  number  is  multiplied  or 
divided  by  10,  100,  ioop,  etc.    <^6,  What  is  a  table  of  logarithms  f 


LOGAKITHMS.  235 

448.  To  Find  the  Logarithm  of  any  Number  from  I  to  10. 

Rule. — Look  for  the  given  numher  m  the  first  line  of  the 
table  ;  its  logarithrn  will  he  found  directly  below  it. 

1.  Find  the  logarithm  of  7.  Ans.  0.84510. 

2.  Find  the  logarithm  of  9.  A7is,  0.95424. 

449.  To   Find  the  Logarithm  of  any  Number  from  10 

to   1000,  inclusive. 

Rule. — Looh  in  the  column  marhed  N"  for  the  first  two 
figures  of  the  given  numher ,  and  for  the  third  at  the  head 
of  one  of  the  other  columns. 

Under  this  third  fiqure*  and  opposite  the  first  tioo,  will 
he  found  the  last  decimal  figures  of  the  logarithm.  The  first 
one  is  found  in  the  column  marked  0. 

To  this  decimal  prefix  the  proper  characteristic.    (Art.  443.) 

Note. — If  the  number  contains  4  or  more  figures,  multiply  the 
tabular  diflference  by  the  remaining  figures,  and  rejecting  from  the 
right  of  the  product  as  many  figures  as  you  multiply  by,  add  the  rest 
to  the  log.  of  the  first  3  figures. 

3.  Find  the  logarithm  of  108.  Ans,  2.03342. 

4.  Find  the  logarithm  of  176.  Ans.  2.2 ^^^\. 

5.  Find  the  logarithm  of  1999.  Ans.  3.30085. 

450.  To  Find  the  Logarithm  of  a  Decimal  Fraction. 
Rule. — Take  out  the  logarithm  of  a  ivhole  number  consist- 

ing  of  the  same  figures^  and  prefix  to  it  the  'proper  negative 
characteristic.     (Art.  444.) 

Note. — If  the  number  consists  of  an  integer  and  a  decimal,  find  the 
logarithm  in  the  same  manner  as  if  all  the  figures  were  integers,  and 
prefix  the  characteristic  which  belongs  to  the  integral  part.  (Art.  443  ) 

6.  What  is  the  log.  of  0.95  ?  Ans.  1.97772. 

7.  What  is  the  log.  of  0.0125?  Ans.  2.09691. 

8.  What  is  the  log.  of  0.0075  ?  ^^^^-  3'375o6. 

9.  What  is  the  log.  of  16.45  ?  Ans.   1.21616. 
10.  What  is  the  log.  of  185.3  ?  Ans.  2.26787. 

448.  How  find  the  logarithm  of  a  numher  from  i  to  10?  449.  From  10  to  1000  ? 
450.  How  find  the  log.  of  a  decimal  ?    Nots.  Of  fta  integer  and  a  decime^l  ? 


236  LOGARITHMS. 

451.  To  Find  the  Number  belonging  to  a  given  Logarithm. 

Rule. — Look  for  the  decimal  figures  of  the  given  logarilhin 
in  the  table  under  the  column  marked  0 ;  a7id  if  all  of  them 
are  7iot  found  in  that  column,  look  in  the  other  colu?nns  on 
the  right  till  you  fi7id  them  exactly,  or  very  nearly  ;  directly 
opposite,  in  the  column  marked  N,  will  he  found  the  first 
iwo  figures,  and  at  the  top,  over  the  logarithm,  the  third 
figure  of  the  given  number. 

Make  this  number  correspond  to  the  characteristic  of  the 
given  logarithm,  by  pointing  off  decimals,  or  by  adding 
ciphers,  if  necessary,  and  it  will  be  the  number  require  L 

Note. — If  the  characteristic  of  a  l(9garithin  is  negative^  the  number 
belonging  to  it  is  2,  fraction,  and  as  many  ciphers  must  be  prefixed  to 
the  number  found  in  the  table,  as  there  are  units  in  the  characteristic 
le%s  I.    (Art.  444.) 

452.  When  the   Decimal  Part  of  the  given   Logarithm  is 
not  exactly,  or  very  nearly,  found  in  the  Table. 

Rule. — From  the  given  logarithm  subtract  the  next  less 
logarithm  found  in  the  tables  ;  annex  ciphers  to  the  remain- 
der, and  divide  it  by  the  tabular  difference  {marked  D) 
as  far  as  necessary. 

To  the  number  belonging  to  the  less  logarithm  annex  the 
quotient,  and  make  the  number  thus  produced  correspond  to 
the  characteristic  of  the  given  logarithm,  as  above. 

Note. — For  every  cipher  annexed  to  the  remainder,  either  a  sig- 
nificant  figure  or  a  cipher  must  be  put  in  the  quotient. 

11.  What  number  belongs  to  2. 1 7231  ?  Ajis.  148.7, 

12.  What  number  belongs  to  1.25 261  ?  A71S.  17.89. 

13.  What  number  belongs  to  3.27715  ?  Jns,  1893. 

14.  What  number  belongs  to  2.30963  ?  Ans.  204. 

15.  What  number  belongs  to  4.29797  ?  Ans.  19858.29. 

16.  What  number  belongs  to  1. 14488  ?  A?is.  0.1396. 

17.  What  number  belongs  to  2.29136  ?  Ans.  0.01956. 

18.  What  number  belongs  to  3.30928  ?  Ans.  0.002038. 

451.  How  find  the  number  belonging  to  a  logarithm? 


LOGARITHMS.  23? 

453.  Computations  by  logarithms  are  based  upon  the 
following  principles : 

1°.  TJie  sum  of  the  logarithms  of  two  numbers  is  equal  to 
the  logarithm  of  their  product. 

Let  a  and  c  denote  any  two  numbers,  m  and  n  their  logarithms, 
and  b  the  base. 

Then  b^  =  a 

And  ft»  =  (J 

Multiplying,  6*+»  =  og, 

2°.  TJie  logarithm  of  the  dividend  diminished  hy  the 
logarithm  of  the  divisor  is  equal  to  the  logarithm  of  the 
quotient  of  the  two  nn7nlers. 

Let  a  and  c  denote  any  two  numbers,  m  and  n  their  logarithms, 


and  b  the  base. 

Then 

fe"*  =  a 

And 

&•  =  <? 

Dividing, 

6«-«  =  a-i-c 

454.  To  Multiply  by  Logarithms. 

1.  Required  the  product  of  35  by  23. 

Solution. — The  log.  of  35  =  1.54407 

«     «'     »  23  =  1.36173 

Adding,  2.90580.    (Art.  453,  Prin.  i.) 

The  number  belonging  is  805,  Ans.     Hence,  the 

Rule — Add  the  logarithms  of  the  factors;  the  sum  ivill 
he  the  logarithm  of  the  product. 

Notes. — i.  If  the  sum  of  tlie  decimal  parts  exceeds  9,  add  the  tens 
figure  to  the  characteristic. 

2.  If  either  or  all  the  characteristics  are  negative,  they  must  be 
added  according  to  Art.  65.  But  as  the  mantissa  is  always  positue, 
that  which  is  carried  from  the  mantissa  to  the  characteristic  must  be 
considered  positive. 

2.  What  is  the  product  of  109.3  by  14.17  ? 

3.  What  is  the  product  of  1.465  by  1.347  ? 

4.  What  is  the  product  of  .074  by  1500  ? 

453.  Upon  what,  two  principles  are  computations  by  logarithms  based?  454.  How 
multiply  by  logarithms  f 


238  LOGARITHMS. 

455.  To  Divide  by  Logarithms. 

5.  Kequired  the  quotient  of  120  by  15. 

Solution.— The  log.  of  120  =       2.07918 
««     4i    «    jg  _        1. 1 7609 

"      **     '*  quotient  =  0.90309.    Ans.  8.    Hence,  the 

Rule.— ^rom  the  logarithm  of  the  dividend  subtract  the 
logarithm  of  the  divisor  ;  the  difference  will  he  the  logarithm 
of  the  quotient.     (Art.  453,  Prin.  2.) 

Notes. — i.  When  either  of  the  characteristics  is  negative,  or  when 
the  lower  one  is  greater  than  the  one  above  it,  change  the  sign  of  the 
subtrahend,  and  proceed  as  in  addition. 

2.  When  I  is  carried  from  the  mantissa  to  the  characteristic,  it 
must  be  considered  positive,  and  be  added  to  the  characteristic  before 
the  sign  is  changed. 

6.  What  is  the  quotient  of  12.48  by  0.16  ? 

7.  What  is  the  quotient  of  .045  by  1.20? 

8.  What  is  the  quotient  of  1.381  by  .096  ? 

456.  Negative  quantities  are  divided  in  the  same  manner 
as  positive  quantities. 

If  the  sign  of  the  divisor  is  the  same  as  that  of  tlie 
dividend,  prefix  the  sign  +  to  the  quotient ;  but  if  different, 
prefix  the  sign  — . 

9.  Divide  —128  by  — 47. 

10.  Divide  — 186  by  — 0.064. 

11.  Divide  — 0.156  by  —0.86. 

12.  Divide  — 0.194  by  0.042. 

457.  To  Involve  a  Number  by  Logarithms. 

Multiplication  by  logarithms  is  performed  by  addition.  (Art.  453.) 
Therefore,  if  the  logarithm  of  a  quantity  is  added  to  itself  once,  the 
result  will  be  the  logarithm  of  the  second  power  of  that  quantity  ;  if 
Added  to  itself  twice,  the  result  will  be  the  third  power  of  that 
quantity,  and  so  on.    Hence,  the 

Rule. — Multiply  the  logarithm  of  the  number  by  the 
exponent  of  the  required  power, 

455.  How  divide  by  them  ?    457  How  involve  a  number  by  logarithms  ? 


LOGARITHMS.  23b 

Notes. — i.  This  rule  depends  upon  the  principle  that  logarithms 
are  the  exponents  of  powers  and  roots,  and  a  power  or  root  is  involved 
by  multiplying  its  index  into  the  index  of  the  power  required. 

2.  In  this  rule,  whatever  is  carried  from  the  mantissa  to  the 
characteristic  is  positive,  whether  the  index  itself  is  positive  or  negative. 

13.  What  is  the  cube  of  1.246. 

Solution. — The  log.  of  1.246  is  0.09551 

Index  of  the  required  power  is  3 

Log.  of  power  is  o.28b53.    Ans.  1.93435. 

14.  What  is  the  fourth  power  of  .135  ? 

15.  What  is  the  tenth  power  of  1.42  ? 

16.  What  is  the  twenty-fifth  power  of  1.234? 

458.  To  Extract  the  Hoot  of  a  Number  by  Logarithms. 

A  quantity  is  resolved  into  any  number  of  equal  factors,  by  dividing 
its  index  into  as  many  equal  parts.    (Art.  281.)    Hence,  the 

Rule. — Divide  the  logarithm  of  the  number  ly  the  index 
oj  the  required  root. 

Note. — This  rule  depends  upon  the  principle  that  the  root  of  a 
quantity  is  found  by  dividing  the  exponent  by  the  number  expressing 
the  required  root.    (Art.  296.) 

17.  What  is  the  square  root  of  1.69? 

Solution.— The  log.  of  1.69  is         0.22789 
The  index  is  2,  2  )  .22789 

Logarithm  of  root,  0.11394.    Ana.  13. 

18.  What  is  the  cube  root  of  143.2  ? 

19.  What  is  the  sixth  root  of  1.62  ? 

20.  What  is  the  eighth  root  of  1549  ? 
:?!.  What  is  the  tenth  root  of  1876  ? 

459.  If  the  characteristic  of  the  logarithm  is  negative, 
and  cannot  be  divided  by  the  index  of  the  required  root 
without  a  remainder,  make  it  positive  by  adding  to  the 
characteristic  such  a  negative  number  as  will  make  it 
exactly  divisible  by  the  divisor,  and  prefix  an  equal  positive 
number  to  the  decimal  part  of  the  logarithm. 

4s8    ■^Tow  extract  the  root  ? 


240  LOGARITHMS. 

22.  It  is  required  to  find  the  cube  root  of  .0164* 
Solution.— The  log.  of  .0164  is  2.21484. 

Preparing  the  log.,  3)3  +  1.21484 

1.40494.    Ans.  0.25406 +. 

23.  What  is  the  sixth  root  of  .001624  ? 

24.  What  is  the  seventh  root  of  .01449  ? 

25.  What  is  the  eighth  root  of  .0001236  ? 

460.  To  Calculate  Compound  Interest  by  Logarithms. 

Rule. — Find  the  amount  of  i  dollar  for  i  year  ;  multiply 
its  logarithm  ly  the  numler  of  years,  and  to  the  product  add 
the  logarithm  of  the  principaL  The  sum  will  be  the  logarithm 
of  the  amount  for  the  given  time. 

From  the  amount  subtract  the  principal,  and  the  remain^ 
der  will  be  the  interest. 

Notes. — i.  If  the  interest  becomes  due  half  yearly  or  quarterly, 
find  the  amount  of  one  dollar  for  the  half  year  or  quarter,  and  multiply 
the  logarithm  by  the  number  of  half  years  or  quarters  in  the  time. 

2.  This  rule  is  based  upon  the  principle  that  the  several  amounts  in 
compound  interest  form  a  geometrical  series,  of  which  the  principal  is 
the  first  term,  the  amount  of  $1  for  i  year  the  ratio,  and  the  number 
of  years  +  i  the  number  of  terms 

26.  "What  is  the  amount  of  1 15 65  for  40  years,  at  6  per 
cent  compound  interest  ? 

Solution.— The  amt.  of  |i  for  i  year  is  $1.06  ;  its  log.,  0.02531 

The  number  of  years,  40 

Product,  1. 01 240 

The  given  principal,  $1565  ;  its  log.,  3-19453 

Ans.  I16103.78.  4.20693 

27.  Wliat  is  the  amount  of  $1500,  at  7  per  cent  compound 
interest,  for  4  years  ?  Ans.  $1966.05. 

28.  What  is  the  amount  of  I370,  at  5  per  cent  compound 
interest  for  33  years  ?  ^/? 5.  $1851.274-. 

460.  How  calculate  compound  interest  by  logarithms  ? 


LOGARITHMS. 


241 


TABLE    OF    COMMON    LOGARITHMS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

.00000 

.3oio3 

.47712 

.60206 

.69897 

.77815 

.84510 

.90309 

.95424 

10 

0432 

0860 

1284 

1703 

2119 

253 1 

2938 

3342 

3743 

4.6 

4i39 

4532 

4922 
8636 

53o8 

569. 

6070 

6446 

6819 

7.88 

7555 

379 

12 

7918 

8279 

ini 

9342 

& 

♦037 

♦38o 

♦721 

1059 

349 

i3 

.11394 

1727 

2057 

2711 

3354 

3672 
6732 

3988 

43o2 

322 

14 

46i3 

4022 
7898 

5229 

5534 

5836 

6137 

6435 

7026 

7319 

3oi 

i5 

7609 

8184 

8469 

8752 

9033 

93i3 

9590 

9866 

♦  140 

281 

i6 

.20412 

o683 

^ll 

1219 

38o5 

1484 

1748 
43o4 

2011 

2272 

253 1 

lin 

264 

\l 

3045 

33oo 

4o55 

455 1 

4797 
7184 

5o42 

249 

7875 

5768 

6007 

6245 

6482 

67.7 

6951 

7416 

7646 

235 

19 

8io3  1  833o 

8556 

8780 

9003 

9226 

9447 

9667 

9885 

223 

20 

.3oio3 

0320 

o535 

0750 

0963 

1175 

1 387 
3445 

1597 

1806 

2oi5 

212 

21 

2222 

2428 

2634 

2838 

3041 

3244 

3646 

3846 

4044 

203 

22 

4242 

4439 

4635 

483 1 

5o25 

5218 

541 1 

56o3 

5704 
7658 

5984 
7840 

\tl 

23 

6173 

636 1 

6549 

6736 
856i 

6922 

7107 
8917 

7291 

7475 

24 

8021 

8202 

8382 

!IS 

9094 

9270 

9445 

9620 

178 

25 

9794 

9967 

♦  140 

♦3l2 

♦654 

♦824 

IX 

1162 

i33o 

171 

i65 

26 

.41497 

1664 

i83o 

1996 

2160 

2325 

2488 

2814 

2975 
456o 

27 

3i36 

3297 

3457 

36i6 

3775 

3933 

tS] 

4248 

44o5 

1 58 

28 

4716 

4871 

5o25 

5179 

5332 

5485 

5788 

5939 

6090 

1 53 

29 

6240 

6389 

6538 

6687 

6835 

6982 

7129 

7276 

7422 

7567 

'47 

3o 

.47712 

7857 

8001 

8144 

8287 

843o 

8572 

8714 

8855 

8996 
♦379 

143 

3i 

9i36 

9276 

9416 

9554 

io55 

9831 

9969 

l322 

♦  106 

♦243 

i38 

32 

.5o5i5 

o65i 

0786 

0920 

1 188 

1455 

1 587 

1720 

1 33 

33 

i85i 

1983 

2114 

2244 

2375 
3656 

25o5 

2634 

2763 

2892 

3020 

i3o 

34 

3i48 

3275 

34o3 

3529 

4778 

3782 

3908 

4o33 

4i58 

4283 

126 

35 

4407 

453 1 

4654 

4900 

5o23 

5i45 

5267 

5388 

6703 

123 

36 

563o 

5751 

5871 

5991 

6110 

6229 

6348 

6467 

6585 

116 

u 

6820 

6937 

7054 

7171 

7287 
8433 

74o3 

7519 

7634 

iin 

7864 
8995 

7978 

8093 

8206 

8320 

8546 

8659 

8771 

n3 

39 

9107 

9218 

9329 

9439 

9550 

9660 

9770 

9879 

9988 

♦097 

no 

40 

.60206 

o3i4 

0423 

o53i 

o638 

Vstt 

o853 

0959 

1066 

1172 

108 

41 

1278 

i384 

1490 

1595 

1700 

1909 

2014 

2118 

2221 

io5 

42 

2325 

2428 

253 1 

2634 

2737 

2839 

2941 

3o43 

3i44 

3246 

102 

43 

3347 

3448 

3548 

3649 

3749 

3849 

3949 

4048 

4147  4247 

100 

44 

4345 

4444 

4542 

4640 

4738 

4836 

5o3i 

5128 

5225 

98 

45 

5321 

5418 

55i4 

56io 

5706 

58oi 

5992 

6087 

6I8I 

95 

46 

6276 

6370 

6464 

6558 

6652 

6745 

6839 

6932 

7025 

7II7 

47 

7210 

73o2 

7394 

7486 

7578 

7669 

7761 

7852 

7943 

8o34 

1 

48 

8124 

8215 

83o5 

8395 

8485 

8574 

8664 

8753 

8842 

8931 
9810 

49 

9020 

9108 

9197 

9285 

9373 

9461 

9548 

9636 

9723 

88 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

u^ 


LOGARITHMS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

B. 

5o 

.69897 

9984 
0842 

♦070 

♦i57 

♦243 

♦329 

♦41 5 

♦5oi 

♦586 

♦672 

86 

5i 

.70757 

0927 

1012 

I933 

1181 

1265 

1 349 

1433 

i5i7 

85 

52 

1600 

1684 

1767 

i85o 

2016 

2099 

2181 

2263 

2346 

83 

53 

2428 

25l0 

2591 

2673 

2754 

2835 

2917 

2997 

3078 

3i59 

8! 

54 

3289 

3320 

3400 

3480 

3560 

1  3640 

IVot 

3799 
4586 

3878 

3957 

80 

55 

■74036 

4ii5 

4194 

4273 

435i 

4429 
52o5 

4663 

4741 

78 

56 

4819 

4896 

4974 

5o5i 

5i28 

5282 

5358 

5435 

55ii 

'll 

^7 

5588 

5664 

5740 

58i5 

5891 

1  5967 

6042 

6118 

6193 
6938 
7670 

6268 

58 

6343 

6418 

6492 

',fj, 

6641 

6716 

6790 

6864 

7012 

75 

59 

7085 

7159 

7232 

7379 

7452 

7525 

7597 

7743 

73 

6o 

.77815 

7887 

7960 

8o32 

8104 

8176 

8247 

83i9 

8390 

8462 

72 

6i 

8533 

8604 

8675 

8746 

8817 

8888 

8958 

9029 

9099 

9169 
9865 

62 

9239 

9309 

9379 

9449 

9519 

9588 

9657 

9727 

9796 

63 

9934 

♦oo3 

♦072 

♦  140 

♦209 

♦277 
0956 
1624 

♦346 

♦414 

♦482 

♦55o 

64 

.80618 

0686 

0754 

0821 

0889 

1023 

n?7 

n58 

1225 

ll 

65 

I2gi 

i358 

1425 

1491 

2l5l 

1 558 

1690 

1823 

1889 
2543 

66 

1954 

2020 

2086 

2866 

2282 

2347 

24i3 

2478 

65 

tl 

2608 

2672 

2737 

2802 

2930 

2995 

3o59 

3i23 

3187 

64 

325i 

33i5 

3378 

3442 

35o6 

3569 

3632 

3696 

3759 

3822 

63 

69 

3885 

3948 

4011 

4073 

4i36 

4199 

4261 

4323 

4386 

4448 

63 

70 

.84510 

4572 

4634 

4696 

4757 

4819 

4881 

4942 

5oo3 

5o65 

62 

71 

5i26 

5.87 

5248 

5309 

5370 

543 1 

5491 

5552 

56i2 

5673 

61 

72 

5733 

5794 

5854 

5914 

5974 

6o34 

6094 

6i53 

6213 

6273 

60 

73 

6332 

6392 

645i 

65io 

6570 
7157 

6629 

6688 

6747 

6806 

6864 

59 

74 

6923 

6982 

7040 

]6T 
8253 

7216 

7274 

7332 

7390 

7448 

5o 

75 

7306 
8081 

7564 

7622 
8196 

7737 
83o9 

ml 

7852 
8423 

7910 

Itl 

8024 

58 

76 

8139 
8705 

8480 

8503 
9154 

57 

77 

8649 

8762 

88r8 

8874 

8930 

8986 

9042 

9098 
9653 

56 

78 

9210 

9265 

9321 

9376 

9432 

9487 

9542 

9598 

9708 

55 

79 

9763 

9818 

9873 

9927 

9982 

♦037 

♦091 

♦  146 

♦200 

♦255 

55 

80 

.90309 

o363 

0417 

0472 

o526 

o58o 

o634 

0687 

074; 

0795 

54 

81 

0849 

0902 

0956 

1009 

1062 

1116 

1 698 

1222 

1275 

i328 

54 

82 

i38i 

1434 

1487 

1 540 

1593 

1645 

1751 

i8o3 

1 856 

52 

83 

1908 

ig6o 

2012 

2o65 

2117 

2169 

2221 

2273 

2324 

2376 

52 

84 

2428 

2480 

253 1 

2583 

2634 

2686 

2737 

2788 

2840 

2891 

52 

85 

2942 

2993 

3o44 

3095 

3i46 

3.97 

3247 

3298 

3349 

3399 

5i 

86 

345o 

3doo 

355i 

36oi 

365i 

3702 

375i 

38o2 

3852 

3902 

5i 

ll 

3952 

4002 

4o52 

41 01 

4i5i 

4201 

425o 

43oo 

435o 

4399 

5o 

4448 

4498 

4547 

4596 

4645 

4694 

4743 

4792 

4841 

4890 

8 

89 

4939 

4988 

5o37 

5o85 

5i34 

5i82 

523 1 

5279 

5328 

5376 

90 

.95424 

5473 

5521 

5569 

56i7 
6095 

5665 

57.3 

5761 

5809 

5856 

48 

91 

5904 

5952 

6000 

6047 

6142 

6190 

6237 

6284 

6332 

47 

92 

6379 

6426 

6473 

6520 

6567 

6614 

6661 

6708 

6755 

6802 

tl 

93 

6848 

6895 
7359 

6942 

6988 

7035 

7081 

7128 

7'74 

7220 

7267 

94 

73.3 

74o5 

745i 

'411 

7543 

7589 

^', 

7681 
8137 
8588 

7727 
8182 

46 

95 

7772 
8227 

78i8 
8272 

7864 
83i8 

ITe^ 

8000 

8046 

45 

96 

8408 

8453 

8498 

8543 

8632 

45 

97 

8677 

8722 

8767 

881 1 

8856 

8901 
9344 

8945 
9388 

8990 

9034 

9078 

45 

98 

9123 

9167 

9211  9255  1 

9300 

9432 

9476 

9520 

44 

99 

9564 

9607 

965 1 

9695 

9739 
4 

9782 

9826 

9870 

9913 

9957 

43 

K. 

0 

1 

2 

3 

5 

6 

7 

8 

9 

D. 

CHAPTER    XX. 

MATHEMATICAL    INDUCTION    AND 
BUSINESS    FORMULAS. 

461.  3Iathematical  Induction  consists  in  proving 
by  trial  that  a  proposition  is  true  in  a  certain  case ;  and, 
finding  it  true  in  the  next  case,  then  in  the  third,  and  so 
on,  we  conclude  it  must  be  true  in  all  similar  cases. 

462.  Many  of  the  principles  and  formulas  of  Arithmetic 
and  Algebra  are  established  by  this  mode  of  reasoning. 

463.  Take  the  familiar  principle  in  Arithmetic: 

The  product  of  any  two  or  more  numlers  is  the  same,  in 
whatever  order  the  factors  are  taken. 

To  prove  this  principle  of  two  numbers,  as  5  and  3,  the  pupil 
represents  the  number  5  by  as  many  unit  marks  in  a 

41,       4P       4t,       4C;       ^ 

horizontal  row,  and  under  this  places  two  similar  rows. 

#  #  #  #  # 
He  sees  that  the  number  of  marks  in  the  horizontal 

#  #  #  #  # 
row  taken  3  times  is  equal  to  the  number  of  marks  in 

a  perpendicular  row  taken  5  times ;  that  is,  3  times  5  =  5  times  3. 

He  then  takes  three  factors  and  finds  the  proposition  true,  and  so 

on.    Hence,  he  concludes  the  principle  is  universally  true. 

464.  Next,  suppose  it  be  asserted : 

The  product  of  the  sum  and  difference  of  two  quantities  is 
equal  to  the  difference  of  their  squares.     (Art.  103.) 
Taking  two  quantities,  as  4  +  3  and  4—3,  or  a  +  &  and  a—T). 
Multiply        4  +  3  Or        a  +  6 

By  4  —  3  By       <g  — & 

42  +  4  X  3  a^  +  ah 

—  4x3  —  3*  —  ab  —  W 

Product,        4^  ^~3^  cf~      -¥ 

461.  In  what  does  Mathematical  induction  consist  ?    Illustration. 


244  MATHEMATICAL     INDUCTION^. 

He  takes  another  example  of  two  quantities,  and  finds  the 
statement  true ;  then  another,  and  so  on.  Hence,  he  concludes  the 
proposition  is  a  universal  truth. 

465.  Suppose  this  proposition  were  enunciated : 

Tlie  sum  of  any  numher  of  terms  of  the  arithmetical  series 
I,  3,  5,  7,  etc,  to  n  terms^  is  equal  to  n^. 

We  see  by  inspection  that  the  sum  of  the  first  two  terms,  1  +  3  —  4, 
or  2- ;  that  the  sum  of  the  first  three  terms,  1  +  3  +  5  =  9>  or  3^ ;  that 
the  sum  of  four  terms,  i  +  3  +  5  +  7  =  i6,  or  4-,  and  so  on.  Hence,  we 
may  conclude  that  the  proposition  is  true  if  the  series  be  extended 
indefinitely.    Or, 

Since  we  know  the  proposition  is  true  when  n  denotes  a  small  num- 
ber of  teims,  and  that  the  value  of  any  term  in  the  series,  as  the  5th, 
7th,  9th,  etc.,  is  equal  to  211—1,  we  may  suppose  for  this  value  of  n, 
that  1  +  3  +  5  +  7 +(2/i— I)  =  w^.  (i) 

Adding  2n  + 1  to  both  members,  we  have 

1  +  3  +  5  +  7  ••••  +(2ri— i)  +  (2?i  +  i)  =  7i2  +  2?i  +  i.  (2) 

Factoring  second  member  of  (2),  7)?  +  2n+i  =  {n+  if. 

Therefore,  since  the  sum  of  n  terms  of  the  series  =  71^,  it  follows 
that  the  sum  of  w  +  i  terms  =  {n+  if,  and  so  on.  Hence,  the  prop- 
osition must  be  universalis/  true. 

466.  In  Geometry,  we  have  the  proposition : 

The  sum  of  the  three  angles  of  a  triangle  is  equal  to  two 
right  ajigles. 

We  find  it  to  be  true  in  one  case;  then  in  another,  etc. 
Hence,  we  conclude  the  proposition  is  universally  true. 

Notes.  — I.  It  is  sometimes  objected  that  this  method  of  f>roof\B  less 
satisfactory  to  the  learner  than  a  more  rigorous  process  of  reasoning. 
But  when  it  is  fully  understood,  it  is  believed  that  it  will  produce  the 
fullest  conviction  of  the  truths  designed  to  be  established. 

2.  In  metaphysics  and  the  natural  sciences,  the  term  induction  is 
applied  to  the  assumption  that  certain  laws  are  general  which  by 
experiment  have  been  proved  to  be  true  in  certain  cases.  But  we 
cannot  be  sure  that  these  laws  hold  for  any  cases  except  those  which 
have  been  examined,  and  can  never  arrive  at  the  conclusion  that  they 
are  necessary  truths. 


BUSINESS     FOEMULAS.  245 


BUSINESS     FORMULAS. 

467.  The  principles  of  Algebra  are  not  confined  to  the 
demonstration  of  theorems  and  the  solution  of  abstruse 
equations.  They  are  equally  applicable  to  the  development 
of  formulas  and  for  business  calculations.  , 

Note. — In  reciting  formulas,  the  student  should  first  state  the 
proposition,  then  write  the  formula  upon  the  blackboard,  explaining 
the  several  steps  by  which  it  is  derived  as  he  proceeds.  He  should 
then  translate  the  formula  from  algebraic  into  common  langv/ige. 

PROFIT    AND    LOSS. 

468.  Profit  and  Loss  are  computed  by  the  principles 
of  Fercenlage. 

469.  To  Find  the  Profit  or  Loss,  the  Cost  and  the  per 

cent  Profit  or  Loss  being  given. 

Let  c  denote  the  cost,  r  the  per  cent  profit  or  loss,  and  p  th* 
percentage,  or  sum  gained  or  lost. 

Since  per  cent  means  hundredths,  r  per  cent  of  a  number  must  be  r 
hundredths  of  that  number.    (Art.  237.)    Therefore, 

c  dollars  x  r  =  p,  the  sum  gained.    Hence,  the 

Formula.       p  =  cr. 
Rule. — MiiUipI^  the  cost  hy  the  rate  per  cent,  and  the 
product  will  he  the  jjrqfit  or  loss,  as  the  case  may  require. 
(Art.  237.) 

1.  Suppose  c  =  $3560,  and  r  =  12  per  cent.  Required 
the  profit. 

Solution.    $3560  x  .12  =  $427.20,  Ans. 

2.  If  a  house  costing  $4370  were  sold  at  8  per  cent  lese 
than  cost,  what  would  be  the  loss  ? 

470.  To  Find  the  per  cent  Profit  or  Loss,  the  Cost  and 

the  Sum  Gained  or  Lost  being  given. 

Let  c  denote  the  cost ;  p  the  percentage,  or  sum  gainea  or  lost ;  and 
r  the  per  cent. 


246  BUSINESS     FOKMTJLAS. 

Since  the  cost  multiplied  by  tlic  rate  gives  ^,  the  given  profit  or 
loss,  it  follows  thatp-f-c  must  be  the  rate.    (Art.  469.)    Hence,  the 

Formula.        r  =  —* 
c 

Rule. — Divide  the  gain  or  loss  hy  the  cost,  and  the  quo- 
tient will  he  the  per  cent  profit  or  loss, 

3.  A  farm  costing  $2500  was  sold  for  $500  advance. 
Required  the  per  cent  profit.  Ans,  20  per  cent. 

4.  A  teacher's  salary  being  $i8oo  a  year,  was  raised  $300. 
What  per  cent  was  the  increase  ? 

471.  To  Find  the  Cost,  the  Profit  or  Loss  and  the  per  cent 

Profit  or  Loss  being  given. 

Let  p  denote  the  sum  gained  or  lost,  r  the  per  cent,  and  c  the  cost. 

Since  the  sum  gained  is  equal  to  the  cost  multiplied  by  the  rate  per 
cent  (Art.  469),  it  follows  that  p  dollars  the  sum  gained,  divided  by  r 
the  rate,  will  be  the  cost.     Hence,  the 

Formula.       c  =  —  • 
r 

Rule. — Divide  the  profit  or  loss  by  the  rate  per  cent. 

5.  The  gain  on  a  bill  of  goods  was  $67.48.  At  25  per 
cent  profit,  what  was  the  cost  ?  Ans.  $269.92. 

6.  An  operator  in  stocks  lost  $1575,  which  was  12I  per 
cent  of  his  investment.     What  was  the  investment? 

472.  To   Find  the  Selling  Price,  the  Cost  and  per  cent 

Profit  or  Loss  being  given. 

Let  c  denote  the  cost,  r  the  per  cent  of  profit  or  loss,  and  s  the 
selling  price. 

When  there  is  a  gain,  the  amount  of  %i  =  i+r;  when  there  is  a 
loss,  the  amount  of  $1  =  i— r ;  and  the  amount  of  c  dollars  =  c{i±  r). 
Therefore,  we  have  the  general 

Formula.        s  =  c{i  ±r). 
Rule. — Multiply  the  cost  hy  i  plus  or  minus  the  rate,  as 
the  case  may  require.     (Art.  240.) 

7.  A  man  paid  $750  for  a  piano.  For  what  must  he  sell 
it  to  gain  15  per  cent?  Ans.  $862.50. 

8.  A  man  bought  a  carriage  for  $960,  and  sold  it  at  a  loss 
of  12 J  per  cent.    What  did  he  receive  for  it  ? 


BUSINESS     FORMULAS.  247 

473.  To  Find  the  Cost,  the  Selling  Price  and  the  per  cent 

Profit  or  Loss  being  given. 

Let  8  denote  the  selling  price,  r  the  per  cent  profit  or  loss,  and  c  the 
cost  required. 

When  there  is  a  gain,  the  selling  price  equals  the  cost  plus  r  per 
cent  of  itself,  that  is,  i+r  times  the  cost;  when  there  is  a  ^o««,  the 
selling  price  equals  the  cost  minus  r  per  cent  of  itself,  that  is,  i— r 
times  the  cost ;  therefore  the  cost  equals  the  selling  price  divided  bj 
I  ±  r.    Hence,  we  have  the  general 

Formula.       c  =  — ; — • 

EuLE. — Divide  the  selling  price  hy  i  plus  or  minus  the 
rate,  as  the  case  way  require  ;  the  quotient  will  he  the  cost. 

9.  A  goldsmith  sold  a  watch  for  $175  and  made  20  per 
cent  profit.    What  was  the  cost  ? 

Solution.  1  +  20  per  cent  =  1.20 ;  and 

$175  -r- 1.20  =  $i45.83i»  ^ns, 

10.  A  jockey  sold  a  horse  for  $540,  and  thereby  lost 
10  per  cent.     What  did  the  horse  cost  him  ? 

SIMPLE    INTEREST. 

474.  The  Elements  or  Factors  involved  in  calcula- 
tions of  interest  are  the  same  as  those  in  percentage,  with  the 
addition  of  time* 

475.  Interest  is  of  two  kinds,  simple  and  compound.  By 
the  former,  interest  is  derived  from  the  principal  only ;  by 
the  latter,  it  is  derived  both  from  the  principal  and  the 
interest  itself,  as  soon  as  it  becomes  due. 

476.  To  Find  the  Time  in  which  any  Sum  at  Simple  Interest 
win  Double  itself,  at  any  given  Rate  Per  Cent. 

Let  p  denote  the  principal,  r  the  rate  per  cent,  i  the  given  interest, 
and  t  the  ti jt  e  in  years. 

Then  i  =  prt,    (Art.  242.) 

Making  i  equal  to  j?,  p  =  prt. 

Dividing  by  pr,  -  ^  t.    Hence,  the 


248  BUSINESS     FORMULAS. 

Formula.       t  —  —. 
r 

'Rule,— Divide  i  by  the  rate,  and  the  q^lotient  will  he  the 
time  required. 

11.  How  long  will  it  take  $1500,  at  5  per  cent,  to  double 
itself  ? 

Solution.    t  =  -  =  —  =  20.    Ans.  20  years. 
r      .05  "^ 

12.  How  long  will  it  take  $680,  at  6  per  cent,  to  double 
itself? 

13.  How  long  will  it  take  I8475,  at  10  per  cent,  to  double 
itself  ? 

477.  To  Find  the  Rate  at  which  any  Principal,  at  Simple 

Interest,  will  Double  itself  in  a  Given  time. 

By  the  preceding  formula,  t  =  -» 

T 

T 

Multiplying  by  - ,  we  have  the 
t 

Formula.        !•  =  -. 

Rule. — Divide  i  hy  the  time  ;  the  quotient  will  he  the  rate 
per  cent  required. 

[For  other  formulas  in  Simple  Interest,  see  Arts.  242-246.] 

14.  If  $1700  doubles  itself  in  8  years,  what  is  the  rate? 

Ans.  12J  per  cent 

15.  At  what  rate  per  cent  will  $5000  double  itself  in 
40  years  ? 

COMPOUND     INTEREST. 

478.  Interest  may  be  compounded  annually,  semi- 
annually, quarterly,  etc.  It  is  understood  to  be  com- 
pounded annually,  unless  otherwise  mentioned. 

479.  To  Find  the  Atnoiitit  of  a  given  Principal  at  Compound 

Interest,  the  Rate  and  Time  being  given* 

Let  p  denote  the  principal,  r  the  rate,  n  the  number  of  years,  ana 
a  the  amount. 


BUSINESS     FORMULAS.  249 

Since  the  amount  equals  the  principal  plus  the  interest,  it  follows 
that  the  amount  of  $i  fori  year  equals  i+r;  therefore,  p(i+r) 
equals  the  amount  of  p  dollars  for  i  year,  which  is  the  principal  for 
the  second  year. 

Again,  the  amount  of  this  new  principal  p{i+r)  for  i  year  = 
p{i+r){i+r)  =  p{i+  rf,  which  is  the  amount  of  p  dollars  for  two 
years. 

In  like  manner,  /)  {\-\-rf  is  the  amount  of  p  dollars  for  three  years 
i,nd  so  on,  forming  a  geometrical  series,  of  which  the  principal  p  is 
the  first  term,  i+r  the  ratio,  and  the  number  of  years  +  i,  the  num- 
ber of  terms.     The  terms  of  the  series  are 

p,    p(i  +  r),    i)(i  +  r)2,    p(n.7.)8^    p(i  +  r)*,  . . . .  i7(i+r>». 

The  last  term,  p  (i  +  Tf»  is  the  amount  of  p  dollars  for  n  years. 
Hence;  the 

Formula.       a  =  p  (i  +  ry. 

Rule. — Multiply  the  principal  by  the  amount  of  %i  for 
I  year,  raised  to  a  power  denoted  by  the  number  of  years ; 
the  product  will  he  the  amount, 

480.  To  Find  the  Compound  Interest  for  the  given  Time 

and  Rate. 

SuUract  the  principal  from,  the  amount,  and  the  remainder 
will  he  the  compound  interest. 

Note. — When  the  number  of  years  or  periods  is  large,  the  oper- 
ations are  shortened  by  using  logarithms. 

1 6.  What  is  the  amount  of  $842,  at  6  per  cent  compound 
interest,  for  4  years  ? 

Solution.    $842  x  (1.06)^  =  $1063,  Am. 

17.  What  is  the  amount  of  $1500,  at  5  per  cent  for  6  yrs., 
compound  interest  ? 

r 

481.  If  the  interest  is  compounded  semi-annually,  -  will 

denote  the  interest  of  $1  for  a  half  year.     Then,  at  com- 
pound interest,  the  amount  of  p  dollars  for  n  years  is 

^\  2n 


'{ 


•+-,) 


'('*f- 


250  BUSINESS     FORMULAS. 

482.  If  the  interest  is   compounded  qiiarte7'ly,  then  - 

4 
will  denote  the   interest   of  $i    for  a  quarter.     Then,  at 

compound  interest,  the  amount  of  jj  dollars  for  n  years  is 

471 

etc. 

4/ 

1 8.  What  is  the  amount  of  $2000,  for  3  years  at  6  per 
fent,  compounded  semi-annually  ?  Ans,  I2388.05. 

19.  What  is  the  amount  of  I5000  for  2  years,  at  4  per  ct., 
compounded  quarterly  ? 

483.  By  transposing,  factoring,  etc.,  the  formula  in 
A.rt.  479,  we  have. 

The  first  term,  «  =  -, • 

The  number  of  terms,  n  =  -^ — ;- — ^~  • 

log.  (I +  7-) 

The  ratio,  r  =  l-r  —  l. 

DISCOUNT. 

484.  Discount  is  an  allowance  made  for  the  payment 
of  money  before  it  is  due. 

485.  The  JPresent  Worth  of  a  debt  payable  at  a  future 
time  is  the  sum  which,  if  put  at  legal  interest,  will  amount 
to  the  debt  in  the  given  time. 

486.  To  Find  the  Present  Worth  of  z  Sum  at  Simple  Interest, 

the  Time  of  Payment  and  the  Rate  being  given. 

Let  8  denote  the  sum  due,  n  the  number  of  years,  and  r  the  interest 
of  $1  for  I  year. 

Since  r  is  the  interest  of  $1  for  i  year,  nr  must  be  the  interest  for  n 
years,  and  i  +7?r  the  amount  of  $1  for  n  years.  Therefore,  s-5-(i  +  nr) 
is  the  present  worth  of  the  given  sum. 

Putting  P  for  present  worth,  we  have  the 

Formula.        B  =  — ; • 

I  4-  fir 

Rule. — Divide  the  sum  due  hy  the  amount  of  $1  for  the 

given  time  and  rate;  the  quotient  loill  he  the  present  worth. 


BUSINESS     FORMULAS.  251 

487.  To  Find  the  Discount,  the  Present  Worth  being  given. 

SuUract  the  present  worth  from  the  debt. 

20.  What  is  the  present  worth  of  $2500  payable  in  4  yrs., 
interest  being  7  per  cent  ?     What  the  discount  ? 

Solution.    P  =  — ?—  =  1^  =  $1953.125,  pres.  worth ,,    , 

i  +  nr        1.28        'a'  V3J     D,  I'  M  ^^^^ 


$2500  —  $1953.125  =    $546,875,  discount. 


1 


21.  What  is  the  present  worth  of  $3600  due  in  5  years,  at 
6  per  cent  ?    WTiat  is  the  discount  ? 

22.  Find  the  present  worth   of  $7800   due  in  6  years, 
interest  5  per  cent  ?    What  the  discount  ? 


COMPOUND    DISCOUNT. 

488.  To  Find  the  Present  Worth  of  a  Sum  at  Compound 
Interest,  the  Time  and  the  Rate  being  given. 

Let  s  denote  the  sum  due,  n  the  number  of  years,  r  the  rate  per  ct. 

Since  r  is  the  rate,  i  +  r  is  the  amount  of  $1  for  i  year ;  then  the 
amount  for  n  years  compound  interest  is  (n-r)«.  (Art.  479.)  That  is, 
$1  is  tlie  present  worth  of  (1+  r)«  due  in  n  years.  Therefore,  «-t-(i  +  r)n 
must  be  the  present  worth  of  the  given  sum. 

Putting  F  for  the  present  worth,  we  have  the 

FOKMULA.  JP  =  V 


(I  +  rf 

Rule. — Divide  the  sum  due  hy  the  amount  of  li  at  com- 
potmd  interest,  for  the  given  time  and  rate;  the  quotient 
will  he  the  present  worth. 

23.  What  is  the  present  worth  of  $1000  due  in  4  years,  at 

5  per  cent  compound  interest  ? 

Solution,    P  =  — ?--  =  f^,  =  $822.71,  Ans. 
(i+r)"     (i.o5)'» 

24.  What  is  the  present  worth  of  $2300  due  in  5  years,  at 

6  per  cent  compound  interest  ? 


252  BUSINESS     FORMULAS. 


COMMERCIAL    DISCOUNT. 

489.  Commercial  Discount  is  a  per  cent  taken 
from  the  face  of  bills,  the  marked  price  of  goods,  etc., 
without  regard  to  time. 

490.  To  Find  the  Cofmnercial  Discount  on  a  Bill  of  Goods, 
the  Face  of  the  Bill  and  the  Per  Cent  Discoant  being  given. 

Let  6  denote  the  base  or  face  of  the  bill,  r  the  rate,  and  d  the 
discount  or  percentage. 

Then  bxr  will  be  the  discount.    Hence,  the ' 

Formula.       d  =  br. 
Rule. — Multiply  the  face  of  the  Mil  hy  the  given  rate,  and 
the  product  will  be  the  cornmercial  discount, 

491.  To  Find  the  Cash  Value  or  Net  Proceeds  of  a  Bill. 

Subtract  the  commercial  discount  from  the  face  of  the  hill. 

25.  Required  discount  and  net  value  of  a  bill  of  goods 
amounting  to  $960,  on  90  days,  at  12^  per  cent  off  for  cash  ? 

Solution.    $q6o  x  .125  =  $120,  discount ;  $960 — $120=1840,  Ans, 

26.  Required  the  cash  value  of  a  bill  amounting  to  I2500, 
the  discount  being  10  per  cent,  and  5  per  cent  off  for  cash. 

492.  To  Mark  Goods  so  as  to  allow  a  Discount,  and  make 

any  proposed  per  cent  Profit. 

Let  c  denote  the  cost,  r  the  per  cent  profit,  and  d  the  per  cent  disc. 

Since  ris  the  per  cent  profit,  1  +r  is  the  selling  price  of  $1  cost,  and 
c{i+t)  the  selling  price  of  c  dollars  cost. 

Again,  since  d  is  the  per  cent  discount  from  the  marked  price,  ano 
the  marked  price  is  100  per  cent  of  itself,  i—d  must  be  the  net  value 
of  $1  marked  price.     Therefore, 

c(i  +7")  -5-  (i— <?)  =  the  marked  price. 

Putting  m  for  the  marked  price,  we  have  the 

Formula.        m  =  -- — -—• 
I  —  d 

Rule. — Multiply  the  cost  hy  i  plus  the  per  cent  gain,  a7id 

divide  the  product  hy  i  minus  the  proposed  discount. 


BUSINESS     FORMULAS.  25,3 

27.  A  trader  paid  I25  for  a  package  of  goods ;  at  what 
price  must  it  be  marked  that  he  may  deduct  5  per  cent,  and 
yet  make  a  profit  of  10  per  cent  ? 

_  c(i+r)     $25x1.10     ^  .  . 

Solution,    m  =  --'^ — ~  =  -^-^ =  $28.94  +  ,  Ans. 

i-d  .95 

28.  A  merchant  buys  a  case  of  silks  at  I1.75  a  yard; 
what  must  he  mark  them  that  he  may  deduct  10  per  cent, 
and  yet  make  20  per  cent  ? 

29.  A  grocer  bought  flour  at  $6^  a  barrel;  what  price 
must  he  mark  it  that  he  may  fall  3  per  cent,  and  leave  a 
profit  of  25  per  cent  ? 


INVESTMENTS. 

493.  The  Value  of  an  Investment  in  National  and 
State  securities,  Railroad  Bonds,  etc.,  depends  upon  their 
market  value,  the  rate  of  interest  they  bear,  and  the  cer- 
tainty of  payment. 

494.  The  Dividends  of  stocks  and  bonds  are  reckoned 
at  a  certain  per  cent  on  the  par  value  of  their  shares,  which 
is  commonly  $100. 

495.  To  Find  the  Per  Cent  which  an  Investment  will  pay,  the 
Cost  of  a  Share  and  the  Rate  of  Dividend  being  given. 

Let  c  denote  the  cost  or  market  value  of  i  sliare  of  stock,  p  \is  par 
value^  and  r  the  annual  rate  of  dividend. 

Since  r  is  the  rate  of  dividend  and  p  the  par  value,  pr  must  be  the 
dividend  on  i  sliare  for  i  year.  Therefore,  pr  -i- c  will  be  the  per  cent 
received  on  the  cost  of  i  share.     (Art.  238.) 

Putting  JR  for  the  per  cent  received  on  the  cost  of  i  share,  we  have 
the 

Formula.        M  =  -— 
c 

KuLE. — Find  the  dividend  on  the  given  shares  at  the  given 
rate,  and  divide  this  hy  the  cost ;  the  quotient  will  he  the  per 
cent  received  on  the  invostment* 


254  BUSINESS     FOKMULAS. 

Note. — When  the  stock  is  above  or  lelow  par,  the  premium  or 
discount  must  be  added  to  or  subtracted  from  its  par  value  to  give  the 
cost. 

30.  What  per  cent  interest  does  a  man  receive  on  an 
investment  of  $5000  in  the  Bank  of  Commerce,  its  dividends 
being  10  per  cent,  and  the  shares  5  per  cent  above  par? 

Solution. — The  premium  on  the  stock  =  $5000  x  .05  =  $2500 
Therefore,  the  cost  =  $5000  +  1250  =  $5250. 

Again,  the  dividend  on  stock  =  $5000 x.  10  =  $5oa  Therefore, 
$5oo-f-$525o  =  9II  per  cent,  Ans. 

31.  A  invested  $6000  in  New  York  6  per  cent  bonds,  at 
3  per  cent  premium.  What  per  cent  did  he  receive  on  his 
investment  ? 

32.  A  man  lays  out  1 1000  in  Alabama  10  per  cents,  at  a 
discount  of  20  per  cent.  What  per  cent  did  he  receive  on 
his  investment  ? 

33.  What  per  cent  will  a  man  receive  on  50  shares  of 
Pennsylvania  Eailroad  stock,  the  premium  being  4  per  ct., 
and  the  dividend  10  per  cent? 

34.  Which  are  preferable,  Massachusetts  6  per  cent  bonds 
at  par,  or  Ohio  8  per  cent  bonds  at  2  per  cent  premium? 

496.  To  Find  the  Amount  of  a  given  Remittance  which  a 
Factor  can  Invest,  and   Reserve  a  Specified    Per  Cent   for    his 
Commission. 

Let  s  denote  the  sum  remitted,  and  r  the  per  cent  commission. 

The  sum  remitted  includes  both  the  sum  invested  and  tlie  commis- 
sion. Now  $1  remitted  is  100  per  cent,  or  once  itself  ;  and  adding  the 
per  cent  to  it,  we  have  i+r,  the  cost  of  $1  invested.  Therefore, 
«  -4-  (i+r)  must  be  the  amount  invested. 

Putting  a  for  the  amount  invested,  we  have  the 

FOEMULA.  a  = 


I+r 


Rule. — Divide  the   remittance  ly  1  plus  the  per  cent 
commission  ;  the  quotient  will  he  the  amount  to  invest. 


BUSINESS     FOEMULAS.  255 

35.  A  clergyman  remitted  to  his  agent  $2500  to  purchase 
books.  After  deducting  4  per  cent  commission,  how  much 
does  he  lay  out  in  books  ? 

Solution,    a  =  — ^  =  ^^^^  =  $2403.85,  Ans. 
i+r        1.04 

;^6.  A  gentleman  remitted  $25000  to  a  broker,  to  be 
invested  in  stocks.  After  deducting  i  J  per  cent,  how  much 
did  he  invest,  and  what  was  his  commission  ? 


SINKING    FUNDS. 

497.  Slnkinf/  Funds  are  sums  of  money  set  apart  or 
deposited  annually,  for  the  payment  of  public  debts,  and  for 
other  purposes. 

CASE    I 

498.  To  Find  the  A^nouitt  of  an  Annual  Deposit  at  Compound 

Interest,  the  Rate  and  Time  being  given* 

Let  s  denote  the  annual  deposit  or  sum  set  apart,  r  the  rate  per 
cent,  n  the  number  of  years,  and  a  the  amount  required. 

Since  the  sam^  sum  is  deposited  at  the  end  of  each  year,  and  put 
at  compound  interest,  it  follows  that  the  deposit  at  the  end  of  the 
ist  year  =  s 
2d     "    =s  +  «(i+r) 
3d     "    =  s  +  s{i  +  r)  +  8{i+rf 
nth.    "     —  8  -\-  s{i+r)  +  s{i+rf  . , . .  +  8{i  ^-r)«-^ 

forming  a  geometrical  series  ;  the  annual  deposit  being  the  first  term, 
the  amount  of  $1  for  i  year  the  ratio,  the  number  of  years  the  num- 
ber of  terms,  and  the  annual  deposit  multiplied  by  the  amount  of  $1 
for  I  year,  raised  to  that  power  whose  index  is  i  less  than  the  number 
of  years,  the  last  term ;  and  the  amount  is  equal  to  the  sum  of  the 
series.    (Art.  402.)    Hence,  we  have  the 

(i  +  ^^r—  I 
FOEMULA.  a  = S. 

r 

Rule. — Multiply  the  amount  of  $i  annual  deposit  for  the 
given  time  and  rate  hy  the  given  annual  deposit ;  the  product 
tvill  he  the  amount  required. 


256  BUSINESS     FORMULAS. 

37.  A  clerk  annually  deposited  $150  in  a  savings  bank 
which  pays  6  per  cent  compound  interest.  What  amount 
will  be  due  him  in  5  years  ? 

Solution,    a  =  ^ -^— ~  a  =     '  ^       $150  =  $845.75.  Ans. 

38.  A  man  agrees  to  give  $300  annually  to  build  a  church 
What  will  his  subscription  amount  to  in  4  years,  at  7  pei 
cent  compound  interest? 

39.  If  a  teacher  lays  up  $500  annually,  and  puts  it  at 
5  per  cent  compound  interest  for  10  years,  how  much  will 
he  be  worth  ? 

CASE    II. 

499.  To  Find  the  Ammal  Dejyosil  required  to  produce  a 
given  Amount  at  Compound  Interest,  the  Rate  and  Time  being  given. 

By  the  formula  in  the  preceding  article,  we  have 

(r  +  r)«  —  I 

^^ s  =  a. 

r 

Dividing  by  coefficient  of  .s,  we  have  the 

FOKMULA.  S=za^  (i  +  vY  —J_ 

r 

Rule. — Divide  the  amount  to  he  raised  hy  the  amount  of 
$1  annual  deposit  for  the  given  time  and  rate  ;  the  quotient 
will  he  the  annual  devosit  rec 


Note. — To  cancel  the  debt  at  maturity,  the  sum  set  apart  as  a 
sinking  fund  is  supposed  to  be  put  at  compound  interest  for  the  giver 
time  and  rate. 

40.  A  father  promises  to  gives  his  daughter  $5000  as  a 
wedding  present.  Suppose  the  event  to  occur  in  5  years, 
what  sum  must  he  annually  deposit  in  a  Trust  Company,  at 
5  per  cent  compound  interest,  to  meet  his  engagement  ? 

(I  +  ^)«  _  I  (105)^  —  I     $250 

Solution.     »  =  a  -*-  L— !— -^ =  $5000  -5-  i — ^ =  — ^  = 

r  .05  .276 

$905.80,  An8. 


BUSINESS     FORMULAS.  257 

41,  A  man  having  lost  his  patrimony  of  $20000,  wishes 
to  know  how  much  must  he  annually  deposited  at  10  per 
cent,  to  recover  it  in  5  years  ? 

42.  A  county  borrows  $30000,  at  6  per  cent  compound 
interest,  to  build  a  court-house ;  what  sum  must  be  set 
apart  annually  as  a  sinking  fund  to  cancel  the  debt  in 
10  years? 

ANNUITIES. 

500.  Annuities  are  sums  of  money  payable  annually, 
or  at  regular  intervals  of  time.  They  are  computed  accord- 
ing to  the  principles  of  compound  interest. 

CASE    I. 

501.  To  Find  the  Amount  of  an  Unpaid  Annuity  at  Compound 

Interest,  the  Time  and  Rate  per  cent  being  given. 

Let  a  be  the  annuity,  1  +  r  tlie  amount  of  $1  for  i  year,  and  n  the 
number  of  years. 

The  amount  due  at  the  end  of  the 

ist  year  =  a, 

2d      *'      =  a  +  a(i+r; 

3d     "     =  a  +  a{i+r)  +  a{i-\-rf, 

4th    "      =   a  +  a{i  +  r)  +  aii^rf  +  a{i+rf, 

nth.  year  =  a  +  a{i  +r)  +  a{i+rf  +  a{i-\rf  . . .  a{i  +r)"-i. 

forming  a  geometrical  progression,  the  annuity  being  the  first  term,  the 
amount  of  $1  for  i  year  the  ratio,  the  number  of  years  the  number  of 
terms,  and  the  annuity  multiplied  by  the  amount  of  $1  for  i  year 
raised  to  that  power  whose  index  is  i  less  than  the  number  of  years, 
the  last  term.  Therefore,  the  amount  is  equal  to  the  sum  of  the 
series. 

Putting  8  for  the  amount  (Art.  498),  we  have  the 

Formula        S  =  (^  +  ^)"  —  i  ^ 

r 

Rule. — Multiply  the  amount  of  %i  annuity,  for  the  given 
time  and  rate,  hy  the  given  annuity. 


258  BUSINESS     FORMULAS. 

Note. — Logarithms  may  be  used  to  advantage  in  some  of  the 
following  examples. 

43.  What  is  due  on  an  annuity  of  $650,  unpaid  for 
4  years,  at  7  per  cent  compound  interest  ? 

Solution.    8  =  (L±j£_Ill «  ^  (L^^ll  $550  =  $2886,  Ans. 
r  .07 

44.  An  annual  pension  of  |88o  was  unpaid  for  6  years ', 
what  did  it  amount  to  at  6  per  cent  compound  interest  ? 

45.  An  annual  tax  of  I340  was  unpaid  for  7  years;  what 
was  due  on  it  at  5  per  cent  compound  interest? 

CASE    II. 

502.  To  Find  the  Present  Worth  of  an  Annuity  atCompound 
Interest,  the  Time  of  Continuance  and  the  Rate  being  given. 

Let  P  denote  the  present  worth ;  then  the  amount  of  P  in  ?i  years 
will  be  equal  to  tlie  amount  of  the  annuity  for  the  same  'time. 
Therefore^ 

r 
Dividing  each  member  by  (i  +  r)"  (Art.  279),  we  have  the 

Formula. 

Note. — In  applying  the  formula,  the  negative  exponent  may  be 
made  positive  by  transferring  the  quantity  which  it  affects  from  the 
numerator  to  the  denominator  (Art.  279). 

1 

r  r 

EuLE. — Multiply  the  present  worth  of  an  annuity  of  $1 
for  the  given  time  by  the  given  annuity. 

46.  What  is  the  present  worth  of  an  annuity  of  $375  for 
6  years,  at  7  per  cent  compound  interest  ? 

Solution.    P  =  l^Jll^ «  =  i-(i.o7H^      =  l=Jt  ^^^^ 
r  .07  .07 

r=  $1785.71,  Ans. 

47.  What  is  the  present  worth  of  an  annual  pension  of 
$525  for  5  years,  at  4  per  cent  compound  interest  ? 


BUSINESS     FORMULAS.  259 

CASE    III. 

503.  To  Find  the  Present  Worth  of  a  Perpetual  Annuity,  the 

Rate  being  given. 

Let  n  denote  infinity,  then  reducing  the  formula  in  Art.  502,  we 
have  this 

Formula.       -P  =  v  C^^*-  435-) 

Rule. — Divide  the  annuity  hy  the  interest  of  %i  for 
I  year,  at  the  given  rate, 

48.  What  is  the  present  worth  of  a  perpetual  scholarship 
that  pays  $150  annually,  at  7  per  cent  compound  interest? 

Solution.    P  =  -  =  i^  =  $2142.86,  Ans. 
r       .07 

49.  "What  is  the  present  worth  of  a  perpetual  ground  rent 
of  $850  a  year,  at  6  per  cent  ? 

CASE    IV. 

504.  To  Find  the  Present  Worth  of  an  Annuity,  commencing 
in  a  given  Number  of  Years,  the  Rate  and  Time  of  Continuance 

being  given. 

Let  n  be  the  number  of  years  before  it  will  cottimence,  and  N  the 
number  of  years  it  is  to  continue.     Then, 

P  —     I  —  (i  +  r)-^"+-^)  __     I  —  (i  +  r)-" 
~  r  r  ' 

Performing  the  subtraction  indicated,  we  have  the 

Formula.        J»  =  -  [(i  +  /»)-«  —  (i  +  r)"^^. 

Rule. — Find  the  present  worth  of  the  given  annuity  to 
the  time  it  terminates  ;  from  this  subtract  its  present  worth 
to  the  time  it  commences. 

50.  What  is  the  present  worth  of  an  annuity  of  I600,  to 
commence  in  4  years  and  to  continue  12  years,  at  7  per 
cent  interest  ? 

SoLxrriON.         P  =  ?  [(i  +  r)-«  —  (i  +  r)-«--»], 

P=?^[(i.o7H-(i.o7)-^«], 
p  ^  $600  X  .4242^^^^^^^^ 


260  BUSINESS     FORMULAS. 

51.  A  father  left  an  annual  rent  of  I2500  to  his  son  for 
6  years,  and  the  reversion  of  it  to  his  daughter  for  12  years. 
What  is  the  present  worth  of  her  legacy  at  6  per  cent 
interest  ? 

CASE  V. 

505.  To  Find  the  Annuity,  the  Present  Worth,  the  Time, 
and  Rate  being  given. 

By  the  formula  in  Article  502, 


p-'- 

(I  +r)-«^ 
r 

Dividing  by  the  coefficient  of  a 

,  we  have  the 

Formula.        a 

JPr 

I  —  (i  +  r)- 

EtiLE. — Divide  the  present  worth  hy  the  present  worth  of 
an  annuity  of%i  for  the  given  time  and  rate. 

52.  The  present  worth  of  a  pension,  to  continue  20  years 

at  6  per  cent  interest,  is  $668.     Eequired  the  pension. 

Pr  $668  X. 06       $668  X. 06    ^  ^        . 

Solution,    a  =  — =  ^^^-7 — -:—r. = ^  ,_, — ■ = $58.23,47^. 

53.  The  present  worth  of  an  annuity,  to  continue  30  years 
at  5  per  cent  interest,  is  $3840.     What  is  the  annuity? 

Note. — The  process  of  constructing  formulas  or  rules,  it  will  he 
seen,  is  based  upon  the  principles  of  generalization  combined  with 
those  of  algebraic  notation.  The  student  will  find  it  a  profitable 
exercise  to  form  others  applicable  to  different  classes  of  problems. 


CHAPTEE    XXI. 

DISCUSSION    OF    PROBLEMS. 

506.  The  Discussion  of  a  Problem  consists  in  assign- 
ing all  tfie  different  values  possible  to  the  arbitrary  quanti- 
ties which  it  contains,  and  interpreting  the  results. 

507.  An  Arbitrary  Quantity  is  one  to  which  any 
value  may  be  assigned  at  pleasure. 

Problem. — If  h  is  subtracted  from  a,  by  what  number 
must  the  remainder  be  multiplied  that  the  product  may  be 
equal  to  c  ? 

Let  X  =  the  number. 

Then  (a  —  ft)  a?  =  c. 

Therefore,  x  = -. 

a  —  o 

508.  The  result  thus  obtained  may  have  five  different 
forms,  depending  on  the  relative  values  of  a,  h,  and  6*.  To 
represent  these  forms,  let  m  denote  the  multiplier. 

I.  Suppose  a  is  greater  than  h.  In  this  case  a  —  6  is  positive, 
and  c  being  positive,  the  quotient  is  positive.  (Art.  112.)  Consequently, 
the  required  multiplier  must  be  positive,  and  the  value  of  x  will  be  of 

the  form  of  +  m. 

« 

II.  Suppose  a  is  less  than  5.  In  this  case  ^  —  &  is  negative^ 
and  c  divided  by  a  —6  is  negative.  (Art.  112.)  Hence,  the  required 
multiplier  must  be  negative,  and  the  value  of  x  is  of  the  form  of  —  m. 

III.  Suppose  a  is  equal  to  h.  In  this  case  a  —  5  =  o.  There 
fore,  the  value  of  x  is  of  the  form  of  — ,  or  a;  =  -  =  00.    (Art.  434.) 

506.  In  what  does  the  discaeeiou  of  problems  consist  ?  507.  What  is  an  arbitrary 
quantity! 


262  DISCUSSION     OF     PROBLEMS. 

IV.  Suppose  c  is  o,  and  a  is  either  greater  or  less  than  h. 

In  this  case  the  value  of  x  has  the  form  — ,  or  a;  =  o. 

m 

V.  Suppose  c  equals  o,  and  a  equals  J.     In  this  case  the 

value  of  a;  has  the  form  -• 
o 

Note. — The  student  can  easily  test  these  principles  by  substituting 
lumbers  for  <z,  &,  and  c. 

509.  The  Discussion  of  Problems  may  be  further  illus- 
trated by  the  solution  of  the  celebrated 


PROBLEM    OF    THE    COURIERS.* 

Two  couriers  A  and  B,  were  traveling  along  the  same 
road  in  the  same  direction,  from  C  toward  Q ;  A  going  at  the 
rate  of  m  miles  an  hour,  and  B  ^  miles  an  hour.  At 
12  o'clock  A  was  at  a  certain  point  P  \  and  B  d  miles  in 
advance  of  A,  in  the  direction  of  Q.  When  and  where  were 
they  together  ? 

0 P  d Q 

This  problem  is  general ;  we  do  not  know  from  the  statement 
whether  the  couriers  were  together  before  or  after  12  o'clock,  nor 
whether  the  place  of  meeting  was  on  the  right  or  the  left  of  P. 

Suppose  the  required  time  to  be  after  12  o'clock.  Then  the  time  after 
12  is  positive,  and  the  time  before  12  is  negative ;  also,  the  distance 
reckoned  from  P  toward  Q  is  positive,  and  from  P  toward  C  is  negntive. 

Let  t  —  time  of  meeting  in  hours  after  12  o'clock  ;  then  mt  =  dis- 
tance from  P  to  the  point  of  meeting. 

Since  A  traveled  at  the  rate  of  m  miles  an  hour,  and  B  n  miles  an 

hour,  we  have 

mt  =  the  distance  A  traveled. 

And  nt-    "        '*        B 

Again,  since  A  and  B  were  d  miles  apart  at  12  o'clock, 

mt  —  nt  =  d. 

Factoring  and  dividing  we  have  the 

*  Originally  proposed  by  Clairaut,  an  eminent  French  mathemati- 
cian, born  in  1713. 


DISCUSSION     OF     PROBLEMS.  263 

Formula.       t  = • 

7n  —  u 

The  problem  may  now  be  discussed  in  relation  to  tlie 
time  t,  and  the  distance  7nf,  the  two  unknown  elements. 

I.  Suppose  m  >  n. 

Upon  this  supposition  the  values  of  t  and  mt  will  both  be  positive; 
because  their  denominator  m  —  n  is  positive.  Now  since  t  is  positive, 
it  is  evident  the  two  couriers  came  together  after  12  o'clock  ;  and  as 
mt  is  positive,  the  point  of  meeting  was  somewhere  on  the  right  of  P. 

These  conclusions  agree  with  each  other,  and  correspond  to  the 
conditions  of  the  problem.  For,  the  supposition  that  m>n  implies 
that  A  was  traveling  faster  than  B.  A  would  therefore  gain  upon  B, 
and  overtake  him  some  time  after  12  o'clock,  and  at  a  point  in  the 
direction  of  Q. 

Let  cZ  =:  24  miles,  m  =  S  miles,  and  n  =  6  miles. 

By  the  formula,    t  = =  ^    ^  =  12  hours. 

m—n       8—6 

mt  =  %  y.  12  —  ()6  miles  A  traveled. 

71?  =  6  X  12  =  72      "     B        *' 

Now,  96  —  72  =  24  m.  their  distance  apart  at  noon,  as  given  above. 

These  values  show  that  the  couriers  were  together  in  12  hours  past 

noon,  or  at  midnight,  and  at  a  point  Q,  96  miles  from  P  and  72  miles 

from  d. 

II.  Suppose  m  <in. 

Then  in  the  formula,  the  denominator  m  —  n\^  negative,  therefore 
both  t  and  mt  are  negative. 

Hence,  both  t  and  mt  must  be  taken  in  a  sense  contrary  to  that 
which  they  had  in  supposition  (I),  where  they  were  positive  ;  that  is, 
the  time  the  couriers  were  together  was  before  12  o'clock,  and  the 
place  of  meeting  on  the  left  of  P. 

This  interpretation  is  also  in  accordance  with  the  conditions  of  the 
problem  under  the  present  supposition.  For,  if  w  <  n,  then  B  was 
traveling  faster  than  A  ;  and  as  B  was  in  advance  of  A  at  12  o'clock, 
he  must  have  passed  A  before  that  time,  somewhere  on  the  left  of  P, 
in  the  direction  of  C. 

Let  (Z  =  24  miles,  m  =  5  miles,  and  ti  =  8  miles. 

By  the  formula,    t  = =  — —  =  —  8  hours. 

m—n       5—8 

.    And  mt  =  5  X  —  8  =  —  40  miles  A  traveled 


264  DISCUSSIOIS"     OF     PKOBLEMS. 

These  values  show  that  the  couriers  were  together  8  hours  before 
noon,  or  at  4  o'clock  a.  m.,  and  at  a  point  C,  40  miles  from  P  and 
64  miles  from  d. 


III.  Suppose  m  =zn. 

Upon  this  supposition  we  have  m  —  n  =  o,  and 


^      d  T  .       md 

t  =  -  =  CO,    also    mt  =  —  =  00. 
o  o 

According  to  these  results,  t  the  time  to  elapse  before  the  couriers  are 
together,  is  infinity  (Art.  434) :  consequently  they  can  never  be  together. 
In  like  manner  mty  the  distance  from  P  of  the  supposed  point  of 
meeting,  is  infinity  ;  hence,  there  can  be  no  such  point. 

This  interpretation  agrees  with  the  supposed  conditions  of  the 
problem.  For,  at  12  o'clock  the  two  couriers  were  d  miles  apart,  and 
it  m  =  n  they  were  traveling  at  equal  rates,  and  therefore  could 
never  meet. 

IV.  Suppose  d  =  o,  and  m  either  greater  or  less  than  n. 

We  then  have  t  = =  o,  and  mt  =  o. 

m  —  n 

That  is,  both  the  time  and  distance  are  nothing.  These  results  show 
that  the  couriers  were  together  at  12  o'clock  at  the  point  P,  and  at  no 
other  time  or  place. 

This  interpretation  is  confirmed  by  the  conditions  of  the  problem. 
For,  if  d  =  0,  then  at  12  o'clock  B  must  have  been  with  A  at  the  point 
P.  And  if  m  is  greater  than  n,  or  m  is  less  than  n,  the  couriers  were 
traveling  at  different  rates,  and  must  either  approach  or  recede  from 
each  other  at  all  times,  except  at  the  moment  of  passing ;  therefore 
they  can  be  together  only  at  a  single  point. 

V.  Suppose  d  =  Of  and  m  =  n. 

Then  we  have  <  =  - »        and        mt  =  -' 

o  o 

These  results  must  be  interpreted  to  mean  that  the  time  and  the 
distance  may  be  anything  whatever,  and  that  the  couriers  must  be 
together  at  all  times,  and  at  any  distance  from  P. 

This  conclusion  also  corresponds  to  the  conditions  of  the  problem. 
For,  if  d5  =  o,  the  couriers  were  together  at  12  o'clock,  and  if  m  =  n, 
they  were  traveling  at  equal  rates,  and  therefore  would  never  part. 


IMAGIJS^ARY     QUANTITIES.  265 


IMAGINARY    QUANTITIESo 

610.  An  Imaginary  Quantity  is  an  indicated  even 
root  of  a  ne^a^m  quantity ;  as,  V— i,  V— «,  ^—7. 

Notes. — i.  Imaginary  quantities  are  a  species  of  radicals,  and  are 
called  imaginary,  because  tliey  denote  operations  wliicli  it  is  impossi- 
ble to  perform.    (Art.  294.) 

2.  Though  the  operations  indicated  are  in  themselves  impossible , 
these  imaginary  expressions  are  often  useful  in  mathematical  analyses, 
and  when  subjected  to  certain  modifications,  lead  to  important  results. 

Sllr  Imaginary  quantities  are  added  and  subtracted  like 
other  radicals.     (Arts.  310,  311.) 

But  to  multiply  and  divide  them,  some  modifications  in 
the  rules  of  radicals  are  required.    (Arts.  312,  313.) 

512.  To  Prepare  an  Imaginary  Quantity  for  Multiplication 
and  Division. 

Rule. — Resolve  tlie  given  quantity  into  two  f actor s,  one 
of  which  is  a  real  quantity,  and  the  other  the  imaginary 
expression  V — 1. 

Notes.— I.  This  modification  is  based  upon  the  principle  that  any 
negative  quantity  may  be  regarded  as  the  product  of  two  quantities, 
one  of  which  is  — i.     Thus,  —a  =  a  x  — i ;  —W  =  ¥  x  — i. 

2.  The  real  factor  is  often  called  the  coefficient  of  the  imaginary 
expression,  -y^— i. 

I.  Multiply  V^^  by  V—'b, 

Solution.  \^—a  =  ^a  x  -v/— i,  and  ^y/— 6  =  '^b  x  /y/— 1« 
Now  y'a  X  -y/^  X  y^  X  y^/^  =  -y/oS  x  — i  =  —y\/ab.  Ana, 


2.  Multiply  +  V--^  by  —  V— y. 

3.  Multiply  V— 9  by  V— ^. 


51a   What    are    Imagimary  quantities?     511.   How  added    and   subtracted  I 
51a  How  prepare  them  for  mxiltiplicatiou  aud  divisiou  1 

1% 


266  IMAGIN^ARY     QUANTITIES. 

513.  It  will  be  seen  from  the  preceding  examples: 

First.  That  the  product  of  two  imaginary  quantities  is  3 
real  quantity. 

Second.  That  the  sign  before  the  product  is  the  opposite 
of  that  required  by  the  common  rule  for  signs.     (Art.  92.) 

For,  while  the  sign  to  be  prefixed  to  an  even  root  is  ambiguous, 
this  ambiguity  is  removed  when  we  know  whether  the  quantity  whose 
root  is  to  be  taken  has  been  produced  from  positive  or  negative 
quantities.    (Art.  293,) 

4.  Multiply  V— 2  by  ViS, 

5.  Multiply  a/— ^  by  \/y. 

Note, — i.  From  these  examples  it  will  be  seen  that  the  product  of 
a  reoyl  quantity  and  an  imaginary  expression,  is  itself  imaginary. 

6.  Divide  V— ^  by  V— ?/. 


Solution.—'^ 1  =  X^-I^v^^  =  V  -,  Ans. 


7.  Divide  V —x  by  V~—x, 

Note. — 2.  Hence,  the  quotient  of  one  imaginary  quantity  divided  by 
another,  is  a  real  quantity  ;  and  the  sign  before  the  radical  is  the  same 
as  that  prescribed  by  the  rule.    (Art.  92.) 

8.  Di^*ide  ^/^x  by  a/.V> 

9.  Divide  Vx  by  V—y, 

Note. — 3.  Hence,  the  quotient  of  an  imaginary  quantity  divided  by 
a  real  one,  is  itself  imaginary,  and  vice  versa. 

10.  Divide  10  a/— 14  by  2  a/— 7, 

1 1.  Divide  c  V'—  i  by  d  V — i, 

514.  The  development  of  the  different  powers  of  V — i. 

12.  (a/^)2  =  -I. 15.  (V^)5  =  +V^. 

13.  (V^^)^  =  -V-i.         16.  {V-if  -  -I. 

14.  {^y~lY=:  +1.  17.    {^~~i)l=  -V'-^. 

Hence,  tJie  even  powers  are  alternately  —  i  and  +1,  and  the  odd 
powers  — /y/— I  and  +y'— i. 


INDETERMIITATE     PROBLEMS,  267 


INDETERMINATE    PROBLEMS. 

515.  An  Indeterminate  Prohlem  is  one  which  does 
not  admit  of  a  definite  answer.     (Art.  220.) 

Note. — Among  the  more  common  indeterminate  problems,  are 

I  St.  Those  whose  conditions  are  satisfied  by  different  values  of  the 

same  unknown  quantity.    (Art  220.) 

2d.  Those  which  produce  identical  equations.     (Art.  200.) 

3d.  Those  which  have  a  less  number  of  independent  simultaneous 

equations  than  there  are  unknown  quantities  to  be  determined. 
4th.  Those  whose  conditions  are  inconsistent  with  each  other. 

1.  Given  the  equation  a;  4-  y  =  9,  to  find  the  value  of  x. 

Solution.  —  Transposing,  x  =  g  —  y,   Ans.     This  result  can   be 
•verified  by  assigning  any  values  to  x  or  y. 

2.  What  number  is  that,  J  of  wliich  minus  i  half  of  itself 
is  equal  to  its  12th  part  plus  its  sixth  part  ? 


Let 

X  = 

the  number. 

Then 

3« 
4 

X  _ 
2  ~~ 

12^6 

Clearing  of  fractions,  etc.. 

gx  = 

gx 

Transposing  and  factoring. 

(9- 

■Ci)X  = 

0 

•• 

.     X  = 

0 
0 

IMPOSSIBLE    PROBLEMS. 

616.  An  Impossible  I^roblemh  one,  the  conditions 
of  which  are  contradictory  or  impossible. 

I.  Given  a;  -f  y  =  10,   x  —  y  =  2,  and  xy  =  38. 

OPERATION. 

Solution. — By  combining  equations   (i)       x  -^  y  =  10      (i) 

and  (2),  we  find  x  =  6  and  y  =  4.    Again,        x y  =  2         (2) 

xxy  =  6x4  =  24.    But  the  third  condition  " 

2X  :i^   I  2 
requires  the  product  of  x  and  y  to  be  38, 

which  is  impossible.  •  *•    ^  =  o 

1=^  4. 

515.  Wb«t  is  au  in4eterifti»ate  problem  ?    516.  Wsat  is  an  impossible  problem  ? 


268  NEGATIVE     SOLUTIONS. 

2.  What  number  is  that  whose  5th  part  exceeds  its  4th 
part  by  15? 

3.  Divide  8  into  two  such  parts  that  their  product  shall 
be  18. 


NEGATIVE    SOLUTIONS. 

517.  A  Neffative  Solution  is  one  whose  result  i&  a 
minus  quantity. 

518.  An  odd  root  of  a  quantity  has  the  same  sign  as  the 
quantity.  An  even  root  of  a  positive  quantity  is  either 
positive  or  negative,  both  being  numerically  the  same. 
(Art.  293.) 

But  the  results  of  problems  in  Simple  Equations,  it  is 
understood,  are  positive;  when  otherwise  it  is  presumed 
there  is  an  error  in  the  data,  which  being  corrected,  the 
result  will  be  positive. 

1.  A  school-room  is  30  feet  long  and  20  feet  wide.  How 
many  feet  must  be  added  to  its  width  that  the  room  may 
contain  510  square  feet  ? 

Solution. — Let  x  =  tlie  number  of  feet, 

Then  (20  +  a;)  30  =  area. 

By  conditions,        600  +  3oaj  =510 
Transposing,  30a;  =  —  90 

.-.     X  —  —  2)  ft.,  Ans. 

Notes. — i.  It  will  be  observed  that  this  is  a  problem  in  Simple 
Equations.  The  steps  in  the  solution  are  legitimate  and  the  result 
satisfies  the  conditions  of  the  problem  algebraically,  but  not  arith- 
metically. Hence,  the  negatke  result  indicates  some  mistake  or 
inconsistency  in  the  conditions  of  the  problem. 

If  we  subtract  3  ft.  from  its  width,  the  result  will  be  a  positive 
quantity. 

2.  Were  it  asked  how  much  must  be  added  to  the  width  that  the 
room  may  contain  690  square  feet,  the  result  would  be  +  3  feet. 

517.  What  is  a  ueeative  eolution? 


hor'N-er's    method.  269 

3.  In  suc«  cases,  by  changing  some  of  the  data,  a  similar  problem 
may  be  easily  found  whose  conditions  are  consistent  with  a  possible 
result. 

2.  What  number  must  be  subtracted  from  5  that  the 
remainder  may  be  8  ? 

Solution.— Let  x  =  the  number. 

Then  5  -  a?  =  8 

Transposing,  a;  =  —  3,  Ans. 

3.  A  man  at  the  time  of  his  marriage  was  36  years  old 
and  liis  wife  20  years.  How  many  years  before  he  was  twice 
as  old  as  his  wife  ?  A71S,  —  4  years. 


HORNER'S  METHOD  OF  APPROXIMATION.^ 

519.  This  method  consists  in  transforming  the  given 
equation  into  another  whose  root  shall  be  less  than  that  of 
the  given  equation  by  the  first  figure  of  the  root,  and 
repeating  the  operation  till  the  desired  approximation  is 
found. 

The  process  may  be  illustrated  in  the  following  manner: 

Let  it  be  required  to  find  the  approximate  value  of  x  in  the  general 
equation, 

A^  +  Bx^  +  Cx  =  D.  (i) 

Having  found  the  first  figure  of  the  root  by  trial,  let  it  be  denoted 
by  a,  the  second  figure  by  6,  the  third  by  c,  and  so  on. 
Substituting  a  for  x  in  equation  (i),  we  have, 

Aa^  +  Ba?  +  Ca  =  B,  nearly. 

Factoring  and  dividing, 

*  So  called  from  the  name  of  its  author,  an  English  mathematician, 
who  communicated  it  to  the  Royal  Society  in  18 19. 


270  hoenek's    method. 

By  putting  y  for  tlie  sum  of  all  the  figures  of  the  root  except  the 
first,  we  have  x  =  a^y,  and  substituting  this  value  for  x  in  equation 
(i),  we  have, 

Aia^yJ  +  B^a  +  yy  +  C{a+y)  =  J); 

or  A  {a^  +  3aV  +  3«/ +2^)  +  ^  («^  +  2ay+y'^)  +  C(a+y)  =  D. 

or     Aa^  +  sAal^y  +  sAay^  +  ^^^  +  Ba'^  +  2Bay  +  5y-  -^-Ca-^Gy —  B. 

Factoring  and  arranging  the  terms  according  to  the  powers  of  ^. 
we  obtain 

Ay^  +  (J5  +  3-4%2  +  ((7+  25a  +  ^Aa})y  =  B-{Ga  +  Ba?  +  Aa^.       (3) 

To  simplify  this  equation,  let  us  denote  the  coefiicient  of  y'^  by  B', 
that  of  ^  by  C",  and  the  second  member  by  D' ;  then, 

Ay^  +  B'f  +  C'2^  =  D\  (4) 

It  will  be  seen  that  equation  (4)  has  the  same  form  as  (i).  It  is  the 
first  transformed  equation,  and  its  root  is  less  by  a  than  the  root  of 
equation  (i). 

By  repeating  the  operation,  a  second  transformed  equation  may  be 
obtained.  Denoting  the  second  figure  in  the  root  by  &,  and  reducing 
as  before,  we  find, 

,  _  D'  . 

C'  +  B'h  +  AJ^'  ^5^ 

Putting  z  for  the  sum  of  all  the  remaining  figures  in  the  root,  we 
have  y  =  !)-{■  z;  and  substituting  this  value  in  equation  (4),  we  obtain 
a  new  equation  of  the  same  general  form,  which  may  be  written, 

As^  +  B"z''  +  C"z  =  D",  (63 

This  process  should  be  continued  till  the  desired  accuracy  is  attained. 
The  first  figure  of  the  root  is  found  by  trial,  the  second  figure  from 
equation  (5),  and  the  remaining  figures  can  be  found  from  similar 
equations. 

But  it  may  be  observed  that  the  second  member  of  equation  {5) 
involves  the  quantity  5,  v.'hose  value  is  sought.  That  is,  the  vahie  of 
6  is  given  in  terms  of  h,  and  that  of  c  would  be  given  in  terms  of  c,  and 
so  on.  For  this  reason,  equations  such  as  (5)  might  appear  at  first 
«ight  to  be  of  little  use  in  practice.  This,  however,  is  not  the  case ; 
for  after  the  root  has  been  found  to  several  decimal  places,  the  value 
of  the  second  and  third  terms,  as  B'b  +  AH^  and  B"c  +  Ac^  in  the 
denominators,  will  be  very  small  compared  with  C  and  G  ',  conse- 


horiter's    method.  271 

quently  as  &  is  very  nearly  equal  to  D'  divided  by  C ,  tliey  may  be 
neglected.  Therefore  the  successive  figures  in  the  root  may  be 
approximately  found  by  dividing  D'  by  C,  D"  by  (7',  and  so  on, 
regarding  C  ,  C ,  etc.,  as  approximate  divisors. 

In  transfonning  equation  (i)  into  (4),  the  second  member  D'  and 
the  coefficients  G  and  B'  of  the  transformed  equation  may  be  thus 
obtained. 

Multiplying  the  first  coeflBcient  A  by  a,  the  first  figure  of  the  root, 
and  adding  the  product  to  B,  the  second  coeflBcient,  we  have, 

B  +  Aa  (7) 

Again,  multiplying  this  expression  by  a,  and  adding  the  product  to 
C,  the  third  coeflicient,  we  have, 

C  +  Ba  +  AaK  (8) 

Finally,  multiplying  these  terms  by  a,  and  subtracting  the  product 
from  B,  we  have 

l)-iCa  +  Ba?  +  Aa^)  =  D', 

which  is  the  same  as  the  expression  for  B'  in  equation  (4). 

Now  to  obtain  C,  we  return  to  the  first  coeflScient,  multiply  it  by  a, 
add  the  product  to  expression  (7),  and  thus  have  the  sum 

B  +  2Aa,  (9), 

which  we  multiply  by  a,  and  adding  the  product  to  expression  (8) 
obtain, 

C  +  2Ba  +  2,Aa^  =  C\ 

which  is  the  desired  coefficient  of  y  in  equation  (4). 

Finally,  to  obtain  B' ,  we  multiply  the  first  coeflBcient  by  a,  and  add 
the  product  to  expression  (9),  and  thus  obtain, 

B  +  sAa  =  B'. 

In  this  way  the  coeflBcients  of  the  first  transformed  equation  are 
discovered  ;  and  by  a  similar  process  the  coeflBcients  of  the  second, 
third,  and  of  all  subsequent  transformed  equations  may  be  found. 

520.  This  metliod  of  approximation  is  applicable  to 
equations  of  every  degree.  For  the  solution  of  cubic  equa- 
tions, it  may  be  summed  up  in  the  following 

KuLE. — 1.  Detach  the  coefficients  of  the  given  equation, 
and  denote  them  hy  A;  B,  (7,  and  the  second  member  hj  D. 
Find  the  first  figure  of  the  root  iy  trial,  and  represent  it  by  a. 


272  HORNER'S     METHOD. 

Multiply  A  ly  a,  and  add  the  product  to  B.  Multiply 
the  sum  hy  a  and  add  the  product  to  C.  Multiply  this  sum 
by  a  and  subtract  the  product  from  D.  The  remainder  is 
the  first  divideyid,  or  D', 

II.  Multiply  A  by  a  and  add  the  product  to  the  last  sum 
under  B.  Multiply  this  sum  hy  a  and  add  the  product  to 
the  last  sum  under  G,  The  result  thus  obtained  is  the  first 
divisor,  or  C. 

III.  Multiply  A  by  a  and  add  the  product  to  the  last  sum 
U7ider  B.     The  result  is  the  second  coefficient,  or  B'. 

IV.  Divide  the  first  dividend  by  the  first  divisor,  TJie 
quotient  is  the  second  figure  of  the  root,  or  b. 

Y.  Proceed  in  like  manner  to  find  the  subsequent  figures 
of  the  root. 

Note. — i.  In  finding  the  second  figure  of  the  root,  some  allowance 
should  be  made  for  the  terms  in  the  divisor  which  are  disregarded  ; 
otherwise  the  quotient  will  furnish  a  result  too  large  to  be  subtracted 
from  ly. 


EXAMPLES. 

I.  Given  a^  +  2x^ 

-{-  Z^  =  24, 

to  find  X, 

SOLXJTION. 

A          B 

C 

D                   a  he 

I               +2 

+  3 

= 

24            a?  =  (  2 .  0  8,  An& 

2 

4 

J 
II 

22 

2  =  jy 

3 

6 

12 

23  =  (7' 

1.891712 

.108288  =  D" 

2 

.6464 

8  =  J5' 

23.6464 

' 

.08 

.6528 

8.08 

24.2992  = 

C" 

..08 

• 

8.16 

.08 

9.24  =  B" 

273 

Note. — 2.  In  the  following  example,  tlie  last  figures  of  tlie  root  are 
found  by  the  contracted  method  of  division  of  decimals,  an  expedient 
which  may  always  be  used  to  advantage  after  a  few  places  of  decimals 
have  been  obtained.    (See  Higher  Arithmetic.) 

2.  Given  ic^  +  izx^  —  i2>x  =  216,  to  find  x. 


SOLUTION. 

A 

B 

0 

D                abe 

I 

+  12 

-18 

=  216          (4.24264+. 

_4 

+64 

184 

x6 

+46 

32  =  iy 

A 

Jo 

26.168 

20 

126  =  0" 

5.832  =  D" 

j4 

4.84 

5.468224 

24  = 

B 

130.84 

.363776  =  2)"' 

24.2 

4.88 

275385 

24.4 
24.6  = 
24.64 
24.68 

=  B" 

135.7  2  =  G' 
.9856 

T36.7  056 
.9872 
I37.0|9|2j8  = 

88391 
82615 

5776 
5508 

X  =  4.24264+,  Ans. 

3.  Given  a:*  +  32:2  +  5a;  =  178,  ^o  find  x. 

a;  =  4.5388,  Ans, 

4.  Given  $3?  +  92:2  —  7^  __  2200,  to  find  x, 

X  =  7.1073536,  Ans. 

5.  Given  a^  +  a^  +  x=  100,  to  find  x, 

a?  =  4.264429+,  Ans. 


274  TEST     EXAMPLES     FOR     REVIEW. 


TEST    EXAMPLES    FOR    REVIEW. 

1.  Required  the  value  of 

6a  +  4^  X  5  +  8«j  -^  2  —  3a  +  12a  X  4. 

2.  Required  the  value  of 

(8a;  +  3^)  5  +  4^  +  7  -  (53J  +  9^)  "^  7- 

3.  Required  the  value  of 

sax  —  ab  +  4cd  —  {2ax  —  4«5  +  2cd), 

4.  Required  the  value  of 

4bc  +  [s^d  —  {2xy  —  mn)  5  +  ^dc], 

5.  Show  that  subtracting  anegative  quantity  is  equivalent 
to  adding  a  positive  one. 

6.  Explain    by  an   example  why  a    positive    quantity 
multiplied  by  a  negative  one  produces  a  negative  quantity  ? 

7.  Explain  why  a  minus  quantity  multiplied  by  a  minus 
quantity  produces  a  positive  quantity. 

8.  Given  ^_(a;  +  8)=^  +  —  -  i7f,  to  find  a;.    . 

3  9        7 

g.  Given  ~ — i h  2a;  =  ^^  X  ^  to  find  x, 

^  5       5  32 

10.  Resolve  ^d^c  —  6Ir^(^  —  c^d  into  two  factors. 

11.  Resolve  $(^y  — -  gxh  —  iSx^yz  into  two  factors. 

1 2.  Resolve  a^'  —  if^'  into  two  factors. 

13.  Resolve  8a  —  4  into  prime  factors. 

14.  Resolve  a^  —  i  into  prime  factors. 

15.  Divide  31  into  two  such  parts  that  5  times  one  of  them 
shall  exceed  9  times  the  other  by  i. 

16.  Make  an  algebraic  formula  by  which  any  two  numbers 
may  be  found,  their  sum  and  difference  being  given. 

17.  Two  sportsmen  at  Creedmoor  shoot  alternately  at  a 
target;  A  hits  the  bull's-eye  2  out  of  3  shots,  and  B  3  out 
of  4  shots;  both  together  hit  it  34  times.  How  many  shots 
did  each  fire  ? 


TEST     EXAMPLES     FOE     REYIEW.  275 

1 8.  Find  two  quantities  the  product  of  which  is  a  and  the 
quotient  b, 

10.  Reduce  -7 7  to  its  lowest  terms. 

^  00  —  0 

20.  Reduce  -^ 75  to  its  lowest  terms. 

c?  —  W' 

21.  Resolve  90:^  _|.  \2xyz  +  42:2  jnto  two  factors. 

22.  Resolve  ()W-  —  die  +  c^  into  two  factors. 

23.  Make  a  formula  by  which  the  width  of  a  rectangular 
surface  may  be  found,  the  area  and  length  being  given  ? 

24.  A  square  tract  of  land  contains  J  as  many  acres  as 
there  are  rods  in  the  fence  inclosing  it.  What  is  the  length 
of  the  fence  ? 

25.  A  student  walked  to  the  top  of  Mt.  Washington  at 
the  rate  of  \\  miles  an  hour,  and  returned  the  same  day  at 
the  rate  of  4^  miles  an  hour ;  the  time  occupied  in  traveling 
being  13  hours.    How  far  did  he  walk? 

26.  Given  l ^^—  =  o,  to  find  x. 

1  —X 

27.  Prove  that  the  product  of  the  sum  and  difference  of 
two  quantities,  is  equal  to  the  difference  of  their  squares. 

28.  Prove  that  the  product  of  the  sum  of  two  quantities 
into  a  third  quantity,  is  equal  to  the  sum  of  their  products. 

29.  Reduce  7-7,  \    "~  ^  —^^ — r  to  its  lowest  terms. 

fj^ J4 

30.  Reduce  j-^ r"^— low-o-ri^v  *o  i*s  lowest  terms. 

31.  Reduce ;; ^  to  a  single  fraction  havirisj 

I  —  a*       I  -\-  a?'  °  *^ 

the  least  common  denominator. 

32.  Find  a  number  to  which  if  its  fourth  and  fifth  part 
be  added,  the  sum  will  exceed  its  sixth  part  by  154. 

33.  Two  persons  had  equal  sums  of  money ;  the  first 
spent  $30,  the  second  I40:  the  former  then  had  twice  as 
much  as  the  latter.    What  sum  did  each  have  at  first  ? 


276  TEST     EXAMPLES     FOR     REVIEW. 

34.  A  French  privateer  discovers  a  ship  24  kilometers 
distant,  sailing  at  the  rate  of  8  kilometers  an  hour,  and 
pursues  her  at  the  rate  of  12  kilometers  an  hour.  How 
long  will  the  chase  last  ? 

35.  Given     '^        ^  =  7   and   7^  — 3^_  ^  ^^^ 

7  2  *^ 

9:  and  y, 

36.  Given  x  =  ^"7     f  5  and  4^ ^!^^  =  3,  to  find 

a;  and  y, 

37.  Make  a  rule  to  find  when  any  two  bodies  moving 
toward  each  other,  will  meet,  the  distance  between  them  and 
the  rate  each  moves  being  given  ? 

38.  A  steamer  whose  speed  in  still  water  is  12  miles  an 
hour,  descended  a  river  whose  velocity  is  4  miles  an  hour, 
and  was  gone  8  hours.     How  far  did  she  go  in  the  trip  ? 

39.  Find  a  fraction  from  which  if  6  be  subtracted  from 
both  its  terms  it  becomes  },  and  if  6  be  added  to  both,  it 
becomes  J. 

40.  Required  two  numbers  whose  sum  is  to  the  less  as  8 
is  to  3,  and  the  difference  of  whose  squares  is  49. 

41.  Given  iox-\-6y  =.  76,  4^—20  =  8,  and  6x-\-Zz=:  88, 
to  find  X,  y,  and  z, 

42.  Given  20;  +  3^  -f  2;  =  24,  ^x  ■\-  y  •\-  2Z  z=z  26,  and 
a;  -f  2y  +  32;  =  34,  to  find  x,  y,  and  z. 

43.  Three  persons,  A,  B,  and  0,  counting  their  money, 
found  they  had  1 180.  B  said  if  his  money  were  taken  from 
the  sum  of  the  other  two,  the  remainder  would  be  |6o; 
C  said  if  his  were  taken  from  the  sum  of  the  other  two,  the 
remainder  would  be  \  of  his  money.  How  much  money 
had  each  ? 

44.  The  fore-wheel  of  a  steam-engine  makes  40  revolutions 
more  than  the  hind-wheel  in  going  240  meters,  and  the 
circumference  of  the  latter  is  3  meters  greater  than  that  of 
the  former.     What  is  the  circumference  of  each  ? 

45.  A  man  has  two  cubical  piles  of  wood;  the  side  of  one 


TEST     EXAMPLES     FOR     REVIEW.  277 

is  two  feet  longer  than  the  side  of  the  other,  and  the  differ- 
ence of  their  contents  is  488  cubic  feet.  Required  the  side 
of  each. 

46.  Required  a  formula  by  which  the  height  of  a  rectan- 
gular solid  may  be  found,  the  contents  and  base  being  given. 

47.  Divide  126  into  two  such  parts  that  one  shall  be  a 
multiple  of  7,  the  other  a  multiple  of  11. 

48.  A  tailor  paid  $1 20  for  French  cloths ;  if  he  had  bought 
8  meters  less  for  the  same  money,  each  meter  would  have 
cost  50  cents  more.     How  many  meters  did  he  buy  ? 

49.  A  shopkeeper  paid  1 175  for  89  meters  of  silk.  At 
what  must  he  sell  it  a  meter  to  make  25  per  cent  ? 

50.  Make  a  formula  to  find  the  commercial  discount,  the 
marked  price  and  the  rate  of  discount  being  given. 

51.  A  man  pays  $100  more  for  his  carriage  than  for  his 
horse,  and  the  price  of  the  former  is  to  that  of  the  latter  as 
the  price  of  the  latter  is  to  50.    What  is  the  price  of  each  ? 

52.  Make  a  formula  to  find  at  what  time  the  hour  and 
minute  hands  of  a  watch  are  together  between  any  two 
consecutive  hours? 

53.  A  father  bequeathed  165  hektars  of  land  to  his  two 
sons,  so  that  the  elder  had  35  hektars  more  than  the  younger. 
How  many  hektars  did  each  receive  ? 

54.  What  number  is  that,  the  triple  of  which  exceeds  40 
by  as  much  as  its  half  is  less  than  5 1  ? 

55.  A  butcher  buys  6  sheep  and  7  lambs  for  $71  ;  and,  at 
the  same  price,  4  sheep  and  8  lambs  for  I64.  What  was  the 
price  of  each  ? 

56.  At  a  certain  election,  1425  persons  voted,  and  the 
successful  candidate  had  a  majority  of  271  votes.  How 
many  voted  for  each  ? 

57.  A's  age  is  double  B's,  and  B's  is  three  times  C's;  the 
sum  of  all  their  ages  is  150.    What  is  the  age  of  each  ? 

58.  Reduce  the  V243  to  its  simplest  form. 

59.  Reduce  Vy^  4-  ay^  to  its  simplest  form. 


278  TEST     EXAMPLES     FOR     REVIEW. 

60.  Reduce  x^  and  y^  to  the  common  index  |^. 

61.  Reduce  ${a  —  h)  to  the  form  of  the  cube  root. 

62.  A  farmer  sold  13  bushels  of  corn  at  a  certain  price  ; 
and  afterward  17  bushels  at  the  same  rate,  when  he  received 
I3.60  more  than  at  the  first  sale.  What  was  the  price  per 
bushel  ? 

63.  A  sold  two  stoves.  On  the  first  he  lost  $8  more  than 
on  the  second;  and  his  whole  loss  was  I2  less  than  triple 
the  amount  lost  on  the  second.  How  much  did  he  lose  on 
each? 

64.  A  number  of  men  had  done  J  of  a  piece  of  work  in 
6  days,  when  12  more  men  were  added,  and  the  job  was 
completed  in  10  days.  How  many  men  were  at  first 
employed  ? 

65.  A  company  discharged  their  bill  at  a  hotel  by  paying 
$8  each;  if  there  had  been  4  more  to  share  in  the  payment, 
they  would  only  have  paid  $7  apiece.  How  many  were 
there  in  the  party  ? 

66.  In  one  factory  8  women  and  6  boys  work  for  $72  a 
week;  and  in  another,  at  the  same  rates,  6  women  and 
1 1  boys  work  for  $80  a  week.  How  much  does  each  receive 
per  week  ?  

67.  What  factor  can  be  removed  from  ViSZX? 

68.  Given  Vx  +  12  =  \/a  +  12,  to  find  x. 

69.  Given  — -  =  ^  ~-    ,  to  find  y. 

y        Vy 

70.  Given  Vz^  —  ^ah  =  a  —  h,  to  find  x. 

71.  From  a  cask  of  molasses  \  of  which  had  leaked  out, 
40  liters  were  drawn,  leaving  the  cask  half  full.  How  many 
liters  did  it  hold  ? 

72.  Make  a  formula  to  find  the  per  cent  commission  a 
factor  receives,  the  amount  invested  and  the  commission 
being  given. 

73.  Divide  20  into  two  parts,  the  squares  of  which  shall 
be  in  the  ratio  of  4  to  9. 


TEST     EXAMPLES     FOR     REVIEW.  279 

74.  After  paying  out  ^  of  my  money  and  then  J  of  the 
remainder,  I  had  I140  left.     How  much  had  I  at  first? 

75.  If  I  be  added  to  both  terms  of  a  fraction,  its  yalue 
will  be  J;  and  if  the  denominator  be  doubled  and  then 
increased  by  2,  the  value  of  the  fraction  will  be  ^.  Kequired 
the  fraction. 

76.  Tiffany  &  Co.  sold  a  gold  watch  for  $171,  and  the  per 
cent  gained  was  equal  to  the  number  of  dollars  the  watch 
cost.     Kequired  the  cost  of  the  watch. 

77.  Two  Chinamen  receive  the  same  sum  for  their  labor; 
but  if  one  had  received  I15  more  and  the  other  $9  less,  then 
one  would  have  had  3  times  as  much  as  the  other.  What 
did  each  receive  ? 

78.  A  drover  bought  a  flock  of  sheep  for  $120,  and  if  he 
had  bought  6  more  for  the  same  sum,  the  price  per  head 
would  have  been  |i  less.  Required  the  number  of  sheep 
and  the  price  of  each. 

79.  A  certain  number  which  has  two  digits  is  equal  to 
9  times  the  sum  of  its  digits,  and  if  63  be  subtracted  from 
the  number,  its  digits  will  be  inverted.  What  is  the 
number  ? 

80.  Two  river-boatmen  at  the  distance  of  150  miles  apart, 
start  to  meet  each  other ;  one  rows  3  miles  while  the  other 
rows  7.    How  far  does  each  go  ? 

81.  A  and  B  buy  farms,  each  paying  $2800.  A  pays  I5  an 
acre  less  than  B,  and  so  gets  10  acres  more  land.  How 
many  acres  does  '^ach  purchase  ? 

82.  Find  a  factor  that  will  rationalize  ^/x  -\-  V7. 

83.  Find  a  factor  that  will  rationalize  V^  —  V^. 

84.  Given  ^W-  +  Vx  z=  -_^-i— ^    to  find  x. 

V{b^  +  Vx) 

85.  The  salaries  of  a  mayor  and  his  clerk  amount  to 
I13200;  the  former  receives  10  times  as  much  as  the  latter. 
Kequired  the  pay  01  eacn. 

S6.  What  two  numbers  are  those  whose  sum  is  to  their 


280  TEST     EXAMPLES     ¥0E     REVIEW. 

difference  as  8  to  6,  and  whose  difference  is  to  their  product 
as  I  to  36  ? 

87.  What  two  numbers  are  those  whose  product  is  48,  and 
the  difference  of  their  cubes  is  to  the  cube  of  their  difference 
as  37  to  I  ? 

S8.  Find  the  price  of  apples  per  dozen,  when  2  less  for 
12  cents  raises  the  price  i  cent  per  dozen. 

89.  Two  pedestrians  set  out  at  the  same  time  from  Troy 
and  New  York,  whose  distance  apart  is  150  miles ;  one  goes 
at  the  rate  of  24  m.  in  3  days,  and  the  other  14  m.  in  2  aays. 
When  will  they  meet  ? 

90.  The  income  of  A  and  B  for  one  month  was  1 187 6, 
and  B's  income  was  3  times  A's.    Required  that  of  each  ? 

91.  A  farmer  bought  a  cow  and  a  horse  for  $250,  paying 
4  times  as  much  for  the  horse  as  for  the  cow.  Find  the 
cost  of  each. 

92.  A  man  rode  24  miles,  going  at  a  certain  rate ;  he  then 
walked  back  at  the  rate  of  3  miles  per  hour  and  consumed 
12  hours  in  making  the  trip.    At  what  rate  did  he  ride  ? 

93.  It  costs  $6000  to  furnish  a  church,  or  |i  for  every 
square  foot  in  its  floor.  How  large  is  the  building,  pro- 
vided the  perimeter  be  320  feet? 

94.  Find  5  arithmetical  means  between  3  and  31. 

95.  Find  the  sum  of  50  terms  of  the  series  -J-,  i,  if,  2,  2 J, 
S,  Zh  4,  4i  etc. 

96.  A  dealer  bought  a  box  of  shoes  for  $100.  He  sold  all 
but  5  pair  for  $135,  at  a  profit  of  li  a  pair.  How  many 
pair  were  there  in  the  box  ? 

97.  Two  numbers  are  to  each  other  as  7  to  9,  and  the 
difference  of  their  squares  is  128.    Eequired  the  numbers. 

98.  In  a  pile  of  scantling  there  are  2400  pieces,  and  the 
number  in  the  length  of  the  pile  exceeds  that  in  the  height 
by  43  :  required  the  number  in  its  height  and  length. 

99.  Bertha  is  J  as  old  as  her  mother,  but  in  20  years  she 
will  be  f  as  old.    What  is  the  age  of  each  ? 

100.  Fifteen  persons  engage  a  car  for  an  excursion ;  but 


TEST     EXAMPLES     FOE     REVIEW.  281 

before  starting  3  of  the  company  decline  going,  by  which 
the  expense  of  each  is  increased  by  $1.75.  What  do  they 
pay  for  the  car  ? 

loi.  When  the  hour  and  minute  hands  of  a  clock  are 
together  between  8  and  9  o'clock,  what  is  the  time  of  day  ? 

102.  A  and  B  wrote  a  book  of  570  pages;  if  A  had 
written  3  times  and  B  5  times  as  much  as  each  actually 
did  write,  they  would  together  have  written  2350  pages. 
How  many  pages  did  each  write  ? 

103.  A  man  and  his  wife  drink  a  pound  of  tea  in  12  days. 
When  the  man  is  absent,  it  lasts  the  woman  30  days.  How 
long  will  it  last  the  man  alone  ? 

104.  Find  the  time  in  which  any  sum  of  money  will 
double  itself  at  7  per  cent  simple  interest. 

105.  A  purse  contains  a  certain  sum,  in  the  proportion  of 
I3  of  gold  to  $2  of  silver ;  if  $24  in  gold  be  added,  there  will 
then  be  $7  of  gold  for  every  $2  of  silver.  Eequired  the  sum 
in  the  purse. 

106.  A  and  B  in  partnership  gain  $3000.  A  owns  f  of 
the  stock,  lacking  $200,  and  gains  $1600.  Required  the 
whole  stock  and  each  man's  share  of  it. 

107.  In  the  choice  of  a  Chief  Magistrate,  369  electoral 
votes  were  cast  for  two  men.  The  successful  candidate 
received  a  majority  of  one  over  his  rival :  how  many  votes 
were  cast  for  each  ? 

108.  Two  ladies  can  do  a  piece  of  sewing  in  16  days;  after 
working  together  4  days,  one  leaves,  and  the  other  finishes 
the  work  alone  in  36  days  more.  How  long  would  it  take 
sach  to  do  the  work  ? 

1 09.  If  a  certain  number  be  divided  by  the  product  of  its 
two  digits,  the  quotient  is  2 J ;  and  if  9  be  added  to  the 
number,  the  digits  will  be  inverted :  what  is  the  number  ? 

no.  Find  4  geometrical  means  between  2  and  486. 

III.  A  trader  bought  a  number  of  hats  for  $80 ;  if  he  had 
bought  4  more  for  the  same  amount,  he  would  have  paid  |i 
less  for  each  :  liow  many  did  he  buy  ? 


282  TEST     EXAMPLES     FOB     KEVIEW. 

112.  If  the  first  term  of  a  geometrical  series  is  2,  the  ratio 
5,  and  the  number  of  terms  12,  what  is  the  last  term? 

113.  A  tree  90  feet  high,  in  falling  broke  into  three 
unequal  parts ;  the  longest  piece  was  5  times  the  shortest, 
and  the  other  was  3  times  the  shortest :  find  the  length  of 
each  piece. 

1 14.  The  sum  of  3  numbers  is  219 ;  the  first  equals  twice 
the  second  increased  by  11,  and  the  second  equals  |  of  the 
remainder  of  the  third  diminished  by  19:  required  the 
numbers. 

1 15.  Required  3  numbers  in  geometrical  progression,  such 
that  their  sum  shall  be  14  and  the  sum  of  their  squares  84. 

1 1 6.  A  pound  of  coffee  lasts  a  man  and  wife  3  weeks,  and 
the  man  alone  4  weeks :  how  long  will  it  last  the  wife  ? 

117.  Two  purses  contain  together  I300.  If  you  take  $30 
from  the  first  and  put  into  the  second,  each  will  then 
contain  the  same  amount :  required  the  sum  in  each  purse. 

118.  A  clothier  sells  a  piece  of  cloth  for  I39  and  in  so 
doing  gains  a  per  cent  equal  to  the  cost.  What  did  he 
pay  for  it? 

119.  A  settler  buys  100  acres  of  land  for  $2450;  for  a 
part  of  the  farm  he  pays  $20  and  for  the  other  part  $30  an 
acre.    How  many  acres  were  there  in  each  part  ? 

120.  What  is  the  sum  of  the  geometrical  series  2,  6,  18, 
54,  etc.,  to  15  terms?  . 

121.  There  are  300  pine  and  hemlock  logs  in  a  mill-pond, 
and  the  square  of  the  number  of  pines  is  to  the  square  of  the 
number  of  hemlocks  as  25  to  49  :  required  the  number  of 
each  kind. 

122.  A  ship  of  war,  on  entering  a  foreign  port,  had 
sufficient  bread  to  last  10  weeks,  allowing  each  man  2  kilo- 
grams a  week.  But  150  of  the  crew  deserted  the  first  night, 
and  it  was  found  that  each  man  could  now  receive  3I  kilo- 
grams a  week  for  the  remainder  of  the  cruise.  What  was 
the  original  number  of  men  ? 


APPENDIX. 


621.    To  Extract  the  Cube  Root  of  Polynomials* 

I.  Required  the  cube  root  of  a^  +  3^5  —  3a*  —  i  la'  -4 
6^8  4.  12a —  8. 

OPERATION. 

a^'+Sa"— a^"*— iia^+6a"2  +  i2a— 8  (  a*+a— 2,  Roots 

a^,  the  first  subtrahend, 
ist  Trial  Divisor,  3a'' )  3a^— 3a**— iia^  etc.,  first  remainder. 

Com.  D.,  2,(1^  +  yi^  +  a- )  3^*^  +  3^^+     d^ 

2d  Tr.  D.,  3a^  +  6«3  4-3a^ )  — 6a*— i2a3  +  6«2  +  i2a— 8,  2d  remainder. 
Complete  Divisor, 

2,a^  +  da^ — 3a'^ — 6a  +  4  )  —  6a*— I2a^  +  6a^  +  i2g— 8.    Hence,  the 

Rule. — I.  Arrange  the  terms  according  to  the  powers  of 
one  of  the  letters^  take  the  cube  root  of  the  first  term  for  the 
first  term  of  the  root,  and  subtract  its  cube  from  the  given 
polynomial. 

II.  Divide  the  first  term  of  the  remainder  hy  three  times 
the  square  of  the  first  term  of  the  root  as  a  trial  divisor,  and 
the  quotient  ivill  be  the  next  term  of  the  root. 

III.  Complete  the  divisor  by  adding  to  it  three  times  the 
product  of  the  first  term  by  the  second,  also  the  square  of  the 
second.  Multiply  the  complete  divisor  by  the  second  term  of 
the  root,  and  subtract  the  product  from  the  remainder, 

IV.  If  there  are  more  than  tioo  terms  in  the  root,  for  the 
second  trial  divisor,  take  three  times  the  square  of  the  part 
of  the  root  already  found,  and  completing  the  divisor  as 
before^  continue  the  operation  until  the  root  of  all  the  terms 
is  found.     ( See  Key) 

*  For  Horner's  Method  of  Approximation,  see  p.  269. 


I. 

x^  ^  gx  +  20. 

2. 

a^  ■}-  ya—  18. 

3- 

a^  —  isa  +  40. 

4. 

2aho^  —  i4aJc  - 

5- 

icy  _2xy  +  I 

6. 

8x^  —  32?/2. 

7. 

a;2  +  yh  +  ?y?7?2;. 

8. 

i2«2a;  —  Sa^y  +  4^2;. 

9. 

a^  —  Sci^x  4-  3a!a;2  —  a^. 

10. 

I  -a^. 

II. 

I  +  Sa\ 

284  APPENDIX. 

2.  Required  the  cube  root  of  a^  +  ^a^  +  30^52  +  b^. 

3.  Find  the  cube  root  of  x^  -\-  6x^  +  122;  +  8. 

4.  Find  the  cube  root  oi  a^  —  6aPy  +  izxy^  —  8y^. 

5.  What  is  the  cube  root  of  Sa^  —  48«r2  _}.  gSa  —  64. 

6.  Find  the  cube  root  of  27^3  —  S4a^x  +  :^6ax^  —  S.'?;^^ 

7.  What  is  the  cube  root  of  a^  —  6a^  -[-  15^*  —  20^3  _j. 
15^2  _  6«  +  I. 

8.  The  cube  root  of  a;^  —  3ic8  ^  Sx^  —  6a^  —  6x^  i-  Sa:^- 
3^+  I. 

522.    Factor  the  following  Polynomials:* 


6odb, 


12.  a^  —  h^x^. 

523.  Find  the  </.  c.  c?.  of  the  following  Polynomials: 

1.  4^2  —  4ax  —  15^  and  6a^  +  'jax  —  3:r2. 

2.  ^ax^yh^,  \2y?^,  and  i6«3a;3;2;2. 

3.  i6a;2  —  2/2,  and  16:^2  _  <^xy  4-  ?/2. 

4.  6^2  ■\-  wax  +3a;2,  and  6^2  +7«a;  —  '^'^^ 

5.  «*  —  5^,  and  a^  —  ^2flr3. 

6.  jr^  —  a^,  and  a;*  —  «*. 

524.  Required  the  h  c.  m.  of  the  following  Polynomials. 

1.  6fl2  _  4fl5,  4058  _^  2«,  and  60^2  _|_  4^, 

2.  4  (i  +  a^),  8(1—  «),  4(1—  «^);  and  8  (i  +  «)• 

3.  c?  —  la  -{•  I,  «'*  —  I,  and  a^  -\-  2a  ■\-  i. 

4.  12  («J2  _  J3)^  4  (^2  _^  ^j)^  and  18  (a2  _  J2). 

5.  4(^2  —  I,  2a  —  I,  and  4^2  -|-  i. 

6.  4  (i  4-  a2),  8(1  +  a),  4  (i  —  a%  and  8  (i  —  «). 

*  The  following  problems  are  classified  and  may  be  studied  in  con- 
nection with  the  subjects  to  which  they  refer,  or  be  omitted  till  the 
other  parts  of  the  book  are  finished,  at  the  option  of  the  teacher. 


APPENDIX.  285 


525.  Unite  the  following  Fractions i 

X  —  2    -  3iP  —  3 

I  2ab 


-  ^3  -  ^   '   ^4  _  ^^ 

ab  be  ac 

5.  From  «  +  3^ take  z(^  —  h-\ ^^« 

2  3 

6.  From  5a;  +  t  take  2:?; 

0  c 

'  X  I 

7.  From  «  +  ic  4-  -^ -^  take  a  —  a;  + 


0^5  —  2^2  a;  4-  ^ 


8.  From take 


a  a  —  I 

I       ■  ,  2 


0.  From  take  -„• 

^  I  —  X  I  —  x^ 

526.  IWultipIy  the  following  Polynomials: 

a  —  b  ,       2b  a^  x^  —  ifi 

1.  I —7  X  2  H ^'  5.  — ; —  X  T^  . 

a  -\-b  a  —  b        X  -^  y  ab 

4a      ^x       2b   ,   s^  ^     o  5^ 

2.  —  +^x h— •  6,  a^  —  I  X  — ^ 

3a;      2^        3a;      4a  a  —  I 

3.  a;2  __  2xy  +  y^  x  —^-^-    7.  -^ ^  x       *^ 


X  —  y  2a  5a  —  10 

75  2^2  —  4<5j3  ^y  ^^. 

527.   Divide  the  following  Quantities: 

,     2a                   2a                3V  2y 

«— 3             «— 3     ^   2y  —  2  y  —  i 

II                     .1          ab  +  b^  b 


X      xy^  '  -^  y     ^'   a^  —  b^   '    a  —  b 

^*  '  "^  2;  •   '       x^'  ^'  \i+a  "^  i-a/  •  (i^a^ 


286  APPEN-DIX. 

528.    Simplify  the  following  Fractions: 

3^  x^ 


2a  — 

-  2 

2a 

a  — 

X  + 

I 

X 

3 

y  + 

4 

.-c^- 

-f 

f       .    . 

I 
a 

-f  ' 
'^  a¥ 

l- 

-^  +  1 

lO. 


529.    Solve  the  following  Equations: 

1.  A  house  and  barn  cost  $850,  and  5  times  the  price  of 
the  house  was  equal  to  12  times  the  price  of  the  barn. 
Required  the  price  of  each. 

2.  A,  B,  and  C,  together  have  145  acres  of  land  ;  A  owns 
two-thirds  and  B  three-fourths  as  much  as  C.  How  many- 
acres  has  each  ? 

3.  From  a  cask  \  full  of  water^  21  liters  leaked  out,  when 
\  the  water  was  left.     Eequired  the  capacity  of  the  cask. 

X  —  4  s^  -j-  14         I 

4.  IX —  4  =  - — — -  —  — 

^  4  3  12 

^3^  +  9        7^  +  5        16  +  40; 


2  3  5 

a;  +  8       a;  —  6 

Y  X  ^=^  X  ■\-  2. 

4  3 

a;  +  8       X  —  6 

X  —  2  ^=z  X  A, • 

4  3 

2a;  -h  I        / —  X  —  3^ 


+  6. 


^-3\ 
4       / 


3 

9.  The  hour  and  minute  hands  of  a  watc^  are  together 
at  12  M.  At  what  time  between  the  hours  of  7  and  8  p.  m. 
will  they  again  be  in  conjunction  ? 

10.  A  merchant  supported  himself  3  yrs.  for  £50  a  year, 
and  at  the  end  of  each  year  added  to  that  part  of  his  stock 
which  was  not  thus  expended,  a  sum  equal  to  \  of  this  part. 
At  the  end  of  the  third  year  his  original  stock  was  doubled. 
What  was  that  stock  ? 


APPENDIX.  287 

530.  Solve  the  following  Simultaneous  Equations,* 

I.    ^^±-'^=26.  3.     ^-^    =9. 

3  •'42^ 

6x       6y  ^  2x  —  2y 

-23  4 

a.    -  +  -  =  8.  4.     -  +  ^  =  d. 

32  23 

X     y  ^  .  y 

2^3  3       4 

5.  A  man  bought  a  horse,  buggy,  and  harness  for  I400 ; 
he  paid  4  times  as  much  for  the  horse  as  for  the  harness, 
and  one-third  as  much  for  the  harness  as  for  the  buggy ; 
how  much  did  he  pay  for  each? 

6.  What  number  consisting  of  two  figures  is  that  to 
which,  if  the  number  formed  by  changing  the  place  of  the 
figures  be  added,  the  sum  will  be  121 ;  but  if  subtracted, 
the  remainder  will  be  9  ? 


X  +  y  —  z  = 

0; 

10. 

xy  =  600 ; 

x+ z—y = 

2; 

xz  =  300  ; 

y  -hz-'X  = 

4. 

yz  =  200. 

II. 

.  y     z 

X   .    z 

«      3      4 

c      d 

a;  ,       ,  z 
-+y  +  -  =  33. 

w  -\-  X  -^  y  = 

:6; 

12 

?^  -f    50  —  ic; 

W  -{■  X  -{-  z  = 

9; 

X  +  120  =  3y; 

IV  +  y  -h  z  = 

8; 

y  -{•  120  =  2Z', 

X  +y  +  z  = 

7- 

^  +  195  =  Z^' 

13.  A's  age  added  to  3  times  B's  and  C's,  is  470  jts.  ; 
B's  added  to  4  times  A's  and  C's,  is  580  yrs. ;  and  C's  added 
to  5  times  A's  and  B's,  is  630  yrs.     What  age  is  each  ? 

14.  What  3  numbers  are  those  whose  sum  is  59  ;  half  the 
difference  of  the  first  and  second  is  5,  and  half  the  differ- 
ence of  the  first  and  third  is  9  ? 


288  APPEl^DIX. 

531.  Generalize  the  following  Problems  and  translate  the  For- 
mulas into  Rules* 

1.  A  dishonest  clerk  absconded,  traveling  5  miles  an 
hour ;  after  6  hours,  a  policeman  pursued  him,  traveling 
8  miles  an  hour.  How  long  did  it  take  the  latter  to  over- 
take the  former  ? 

Note, — Substitute  c  for  clerk's  rate,  p  for  policeman's  rate,  n  foi 
flumber  of  hours  between  starting,  and  x  for  the  time  required. 

Formula.       x  = • 

p  —  c 

2.  A  can  do  a  piece  of  work  in  2  days,  B  in  5  days,  and 
0  in  10  days ;  how  long  will  it  take  all  working  together 
to  doit? 

Note. — Let  «,  &,  and  c  represent  the  numbers.  Then  x  =  dbe  -*• 
{ctb  +  ac  +  be). 

-r,  abc 

Formula.       x  =  — ; ^-« 

ab  +  ac  +  be 

3.  Divide  $4400  among  A,  B,  and  C,  in  proportion  to  the 

numbers  5,  7,  and  10. 

Note.— Put  a,  b,  and  c,  for  the  proportions,  s  for  the  sum  of  the 
proportions,  and  n  for  the  number  to  be  divided. 

4.  A  father  is  now  9  times  as  old  as  his  son ;  9  years 
hence  he  will  be  only  3  times  as  old :  what  is  the  age 
of  each? 

632.   Expand  the  following  by  the  Binomial  Theorem  : 

1.  {2a--2fiY.  4.     (^'  +  /)'. 

2.  {zx  +  2y)\  5.    {a  +  a-^)\ 

3.  (i  +  3«)'.  6.    (fl2  _  2a)\ 
533.   Find  the  Product  of  the  following  Powers  : 

1.  dbor^  by  a^.  3.    x~'^  by  x~^, 

2.  a^h-^x-^}ijarWx-\  4.    y-'^hjy\ 

634.  Divide  the  following  Powers: 

1.  6a~»  by  3«-2.  3.     1 2x-^  by  4X~^. 

2.  Sa-*bc-^  by  4a^¥c^,  4.    {a + a;)-»  by  {a  +  a;)-»». 


APPEi^^DIX.  289 

535.  Transfer   Denominators   to    Numerators,  thus   forming 
entire  Quantities. 

J      ^,  ^,     -^ — • 

536.  Unite  the  following  Radicals : 

1.  V48  +  V27  +  '\/243.        4.  X  A/25^  +  "^/z^xf^c, 

2.  A/54^  —  '\/9(>x  4-  A/24a;.   5.  V^ofi^^  —  '\/2oa^x. 

3.  8  V^^^jJ  +  2  Vi^.  6.  3'V^i28iz:3z/';2  —•  ^x  \^i6yz. 

537.  Find  the  Product  of  the  following  Madicals: 

1.  («  4-  ?/)i  X  (^  +  70^.      3-     (^'  +  y)^  X  (x  H-  2/)i 

2.  4  +  2^/2x2  —  \/2.      4-  3^  V^^  +  ^  X  4  a/^. 

538.  I>ividing  one  Radical  by  Another. 

1.  (fl^%)-  -^  («a;)^.  4.     (^  +  yf  -J-  (^  +  2^)". 

2.  24a;  ^ay  -^  6  V^.  5.     4«  a/«^  -i-  2  V«c- 

3.  "s/ i6a^  —  \2aH -^  2a.        6.     yoyp-i-yViS. 

539.  Required  the  Factors  which  will  Rationalize  the  fol- 
lowing Radicals: 

1.  2  ^/a  +  V7.  4.     \/5  —  \/5. 

2.  ic  +  Vy.  5-     4  V2S  —  5  Vy. 

3.  Thedenom.  of-^.     6.     The  d.  of -— -^-= 

2V3  V3+V2+I 

540.  Solve  the  following  Madical  Equations: 

1.  Given  Vx  +  i  =  Vn  +  ^j  to  find  x. 

2.  Given  \/x  +  18  —  ^5  =  V^  —  75  to  find  x, 

3.  Given  Vx^  —11=5.     5.     (13  +  ^23  +  y"^)^  =  5. 

6  _  

4-        /--— =^  =  V  3  +  ^-       6.     2  V^  =  V  iz^  +  3«. 


290  APPEKDIX. 

541.  Solve  the  following  Quadratics, 


2.      — 


0-^  - 

^~3 

"~~  ' 

..    J         ^ 

i6 

X 

lOO  — 

4^2 

gx  ^ 

=  3. 

^ 

a^ 

I 

2 

4 

32 

Vi^ 

L±_?  _ 

4  — 

V5 

4  -(-  Va;  \^x 

5.  Find  two  numbers  whose  differenco  is  12,  and  the  suni 
of  their  squares  1424. 

6.  Eequired  two  numbers  whose  sum  is  6,  and  the  sum 
of  their  cubes  7  2. 

7.  Divide  the  number  56  into  two  such  parts,  that  their 
product  shall  be  640. 

8.  A  and  B  started  together  for  a  place  150  miles  distant. 
A's  hourly  progress  was  3  miles  more  than  B's,  and  he 
arrived  at  his  journey's  end  8  hrs.  20  min.  before  B.  What 
was  the  hourly  progress  of  each  ? 

9.  The  diiference  of  two  numbers  is  6 ;  and  if  47  be 
added  to  twice  the  square  of  the  less,  it  will  be  equal  to  the 
square  of  the  greater.     What  are  the  numbers  ? 

10.  The  length  added  to  the  breadth  of  a  rectangular 
room  makes  42  feet,  and  the  room  contains  432  square  feet. 
Required  the  length  and  breadth. 

11.  A  says  to  B,  the  product  of  our  years  is  120 ;  if  I 
were  3  yrs.  younger  and  you  were  2  jrrs.  older,  the  product 
of  our  ages  would  still  be  120?    How  old  is  each  ? 

12.  Vx^  +  a/^  =  6  Vx. 

13.  X  4-  '\/x  +  6  =  2  +  3  a/^  +  6. 

14.  A  man  bought  80  lbs.  of  pepper  and  100  lbs.  of 
ginger  for  £65,  at  such  prices  that  he  obtained  60  lbs.  more 
of  ginger  for  £20  than  he  did  of  pepper  for  £10.  Whai 
did  he  pay  per  pound  for  each  ? 


COLLEGE  EXAMINATION  PROBLEMS. 

542.  I.  Divide  i^x^  —  ^~  hjx 

5<?  c 

2.  Divide  a*  —  ¥  hy  {a  —  b), 

3.  Solve  tne  equation  a;  + =  12  —  -^ — • 

4.  Multiply  3  V45  —  7  V7  ^y  Vif  +  2  Vpf 

5.  Divide  a^b^  by  «^5i     6.  Divide  xy~^  by  a;%~^, 

7.  Given  ^cd^  -[-  2X  —  9  =  76,  to  find  x. 

8.  Given  ia;^  _  t^;  +  7I  =  8,  to  find  x. 

9.  Find  two  numbers,  the  greater  of  which  shall  bw  to 
the  less  as  their  sum  to  42,  and  as  their  difference  to  6. 

10.  Find  the  value  of  i  +  |  +  ^f  +  ^f  +  ^^^*  ^^  infinity. 

11.  Find  the  third  term  of  {a  +  by^. 

12.  Expand  to  four  terms  (i  +  x^)~k 

543.  I.  Divide        .    ^  +  -  by  — ' — ^ ; 


2.  Find  the  product  of  a^,  a\  a^  and  a  ^^. 


2a 


Va^  +  ^ 


3.  Solve  the  equation  x  +  V«^  +  ^^ 

4.  Solve  ^^ ^1 -^—  =  o. 

a;       X  -\-  I       X  -{■  2 

5.  Solve  V^^  I  =  a?  —  !• 

6.  Find  the  value  of  f  +  i  +  J  +,  etc.,  to  infinity. 

7.  Given  x-{-'\/x  :  x—Vx  ::  3  ^^  +  6  :  2  V^,  to  find  x, 

8.  Expand  to  four  terms  (a^  +  x)'^. 

9.  Expand  to  four  terms  {x^  —  y^)~^. 

544.    I.  Find  the  g.  c.  d.  and  the  I.  c.  m.  of  (243aioj5  _^  ^  j 
and  {^ia%^  —  i)  by  factoring. 

T^.  .,       6  V^   1       20c  Vl^ 

2.  Divide  ^— r  by  ^— • 

25  v«^        2iaJ  wd^ 

3.  Solve  the  equations  2X  —  y  —  21,  20^2  +  ^/^  =  153. 


292  APPENDIX. 

4.  A  person  buys  cloth  for  $90.  If  lie  had  got  two  yards 
more  for  the  same  sum,  the  price  would  have  been  50  cts. 
per  yd.  less.     How  much  did  he  buy,  and  at  what  price  ? 

5.  Expand  (a  —  b)^  by  the  binomial  theorem. 

6.  Factor  4^^  —  ^y\ 

7.  Multiply  3  Y  ?  by  2  W  |- 

8.  Given  x  +  2y  =  y  and  22:4-3^  =  12,  to  find  z  and  y, 

9.  Eeduce  a  V4Sa^d  and  Vj|-  to  their  simplest  forms. 

10.  Given 4-  -  =  12 ,  to  find  x. 

32  3     ^ 

X  or  — —  rt' 

645.    I.  From  ^x  ■{ — =  subtract  x  — 


2h  c 

2.  Multiply  together  -^^ — ,    ~~~2>  ^^^  ^  "^ ~ — * 

I  -J-  y     X  -f-  X  I  —  X 

3.  Extract  the  square  root  of  Sal)^  +  a*  —  4a^  -f-  4 J*. 

4.  From  2  V3I0  take  3  'V^4o. 

5.  Divide  a"^^^  by  a^b~^.      6.  Solve  a;*  +  40^2  —  12. 

7.  Solve  a;2  —  X  Vs  =  x  —  ^  \/^. 

8.  What  two  numbers  are  those  whose  sum  is  2a,  and  the 
sum  of  their  squares  is  2^*  ? 

9.  What  two  numbers  are  those  whose  difference,  sum, 
and  product  are  as  the  numbers  2,  3,  and  5  respectively  ? 

10.  Find  three  geometrical  means  between  2  and  162. 

11.  Expand  to  four  terms 


546.    T.  Divide  120;^  —  192  by  ^x  —  6. 

2.  Divide  a;*  H 7  by  7  —  m, 

3.  Solve  the  equation  21  +  ^-^^  =  5^  +  2Zzi2?. 

^  16  82 

4.  Find  the  product  of  a^,  «^,  «^,  and  «~^. 

5.  Solve  the  equation  ~  — g—  =  2. 

3/  2X 

6.  Find  6  arithmetical  means  between  i  and  50. 


COLLEGE     PEOBLEMS.  293 

7.  How  many  different  combinations  may  be  formed  of 
eight  letters  taken  four  at  a  time  ? 

8.  Expand  {a  —  l)~^  to  four  terms. 

9.  Divide  150  into  two  such  parts  that  the  smaller  may 
be  to  the  greater,  as  7  to  8. 

10.  Given  5a;  +  2?/  =  29  and  2^  —  a;  =  —  i,  to  find 
X  and  y, 

2  T 

547.   I.  Solve  the  equation {-  4  =  a 

2.  What  is  the  relation  between  a,  ap,  and  ar^  ? 

3.  Find  two  numbers  such  that  the  sum  of  |  the  first 
and  \  of  the  second  equals  1 1,  and  also  equals  three  times 
the  first  diminished  by  the  second. 

4.  Give  the  first  three  and  last  three  terms  of  (2« )  • 

5.  Find  the  r/.  c.  d,  of  a^  —  W  and  a^  —  lab  +  V^, 

6.  Find  the  I,  c.  in,  of  (p?  —  ^),  and  4  (a  —  x\  and 

7.  Add  -=^-^ f ,  —  ^ =:r,  and  ^—, tto* 

5  (a  —  ^  5  («  —  *)  5_(«  —  ^)^ 

8.  Find  the  value  of  x  in  "^x  +  a  =  V^  +  a. 

648.    I.  Eeduce  r 5 to  its  lowest  terms. 

c?  —  x^ 

2.  Multiply  ar^m  by  -^  ;  and  divide  ar^W  by  -^ 

o  1      J.1  .  •       nx  —  6         X  —  5  <^ 

3.  Solve  the  equation 7 ^^—  =  — 

^  35  6;r  —  loi       s 

4.Give.Z£±J_(._£^)  =  ,. 

a^      X 

5.  Solve  the  equation  -  -^ [-  7I  =  8. 

6.  It  is  required  to  find  three  numbers  such  that  the 
product  of  the  first  and  second  may  be  15,  the  product  of 
the  first  and  third  21,  and  the  sum  of  the  squares  of  the 
second  and  third  74. 


29":^  COLLEGE     PROBLEMS. 

7.  Find  tlie  sum  of  n  terms  of  the  series  i,  2,  3,  4, 
5^  6,  etc. 

8.  Expand  to  five  terms  {cfi  —  ^3)-^. 

9.  Find  the  sum  of  the  radicals  A/300  and  a/tJ. 
10.  Solve  the  quadratic  -  -| =  2^, 

549.  I.  Find  the  sum  and  difference  of  \/i8«^and 

2.  Multiply  2  Vi  -  a/3^  by  4  V3  -  2  a/^. 

3.  Solve  the  equation      """  -  4-  ^^  ~~  ""^  z=z  n  ^  ^  -r  ^^ 

7  5^4 

4.  Solve  the  equation      ~~  ^ ^~^  =  — . 

a;  —  2      X  —  I       20 

5.  The  sum  of  an  arithmetical  progression  is  198 ;  its 
first  term  is  2  and  last  term  42  ;  find  the  common  differ- 
ence and  the  number  of  terms. 

6.  Expand  to  four  terms  {a^  —  ]/)^, 

7.  Simplify  the  radical  (cfi  —  20^!)  -\-  aW)^, 

8.  A  and  B  together  can  do  a  piece  of  work  in  3I  days, 
B  and  0  in  4f  days,  and  0  and  A  in  6  days.  Required  the 
time  in  which  either  can  do  it  alone,  and  all  together. 

9.  Find  3  numbers  such  that  the  prod,  of  the  first  and 
second  may  be  15,  the  prod,  of  the  first  and  third  21,  and 
the  sum  of  the  squares  of  the  second  and  third  74. 

550.  I.  Given  -  +  -==2,  -  +  -  =  3,  --{--  =  3;  find 
X,  y,  and  z. 

2.  Given  ^ =: ,  to  find  x. 

S  —  X  3  12 

3.  Find  the  I.  c.  m.  and  g.  c.  d.  oi  x^  ■\'  ^  —  21  and 
x^  —  X  —  $6, 

4.  It  takes  A  10  days  longer  to  do  a  piece  of  work  than 
it  takes  B,  and  both  together  can  do  it  in  12  days.  In  how 
many  days  can  each  do  it  alone  ? 

5.  Substitute  ?/  +  3  f or  a;  in  >*  —  a?  -{■  2X^  —  t,  \  simplify 
and  arrange  the  result. 


ANSWERS. 


Page  15. 

I,  2.  Given. 

3.  2  cts.  A,  6  cts.  0. 

4.  $8,  h. ;  I32,  c. 

5.  9  and  27. 

6.  4^,0;  8^,  B;  16^?,  A 

7.  121/,  son  ;  z^y>  father. 

Pagre  16* 

8.  $20,  B's ;  I80,  A's. 

9.  15.  30.  45- 

10.  $7,  calf;  $56,  cow. 

II.  $5.25,  bridle; 
$10.50,  saddle; 
$110.25,  horse. 

12.  $3000,  daughter; 

I6000,  son ; 

$27000,  wife. 
13-   234,  702,  936. 

Page  19. 

1-3.  Given. 

4.  98^. 

5.  18. 


INTRODUCTION. 

6.    10. 
7-  34- 


8.   17. 

Page  21c 

1.  60. 

2.  40. 

3.  ac  +  85. 

4.  55  —  2t7. 

5-  35- 

6.  24. 

7.  3^  +  21/  +  «5. 

8.  6b  —  jcx  +  sa, 

9.  Z>a;^  +  ca:?/. 

Page  22, 

10.  -^-^  +  a. 

2Z 

b  —  a  , 

II. f-  2;^. 

12.  3^  +  i?^«/  +  %2;. 

^3- d 

'  14.  92. 
15.   120. 


ax  —  Qty  —  Zia;  +  by 


29G 


S  U  B  T  K  A  C  T I  0  If . 


ADDITION. 


Page  24, 

l^wgre  26. 

1,  2.  Given. 

I.   240^  +  25—36?. 

3.  2iah. 

2.  iGmn—xy-^-hc 

4.   11  xy. 

3.   i6hc -\- xy -—mn 

5.   isa\ 

4.  4fl!5— 3w/^  +  2^ 

6.   —  22,hcd. 

5.   i52:?/-i-«5  +  5. 

7.    —  i6:r3|/2. 

6.  Given. 

8.  45a^2. 

7.  21  {a  +  h). 

9.   -39«^^y. 

8.   igc{x—y). 

10.   2gWd7n\ 

9.  7«V^y. 

II.  Given. 

10.  6^/^?. 

12.  4. 

II.   loVa; — y. 

13.  5. 

Page  27. 

12.  Given. 

J*asre  ;^5. 

13.  a{T~6h  +  ^d 

14,  15-  Griven. 

—3>m). 

16.  82;. 

14.  2/(^^  +  3  — 2C 

17.  «^c. 

-5m). 

18.   —  12J. 

15.  m{g  +  ab--jc 

19.   —  12?/. 

+  3^). 

20.   —  2m. 

16.    i?;(l3«r_3^_|_^ 

21.   I. 

-3^+?^). 

22.  75- 

17.  a;?/(a4-5-c). 

23.  Given. 

I.  Given. 

4. 

5- 
6. 

7. 
8. 

9- 
10. 
II. 
12. 

13. 

14. 

15- 
16. 

17. 

18. 
19. 
20. 


.   16  cts.,  b ; 
30  cts.,  k. 

Page  28, 

26  peaches ; 
49  pears. 
15  and  70. 
15  K  25  g. 
I. 

7. 

7. 

356,  A;  94,  B. 

36  and  141. 

8. 

9  cts.,  top; 

23  cts.,  ball. 

$9?  b;  I31,  s. 

26  cts.,  A.  M.; 

74  cts.,  p.  M. 

Given. 
14. 

7. 
12. 
60. 
20. 


I'cffire  31, 

I,  2.  Given. 

3.  i4^yz. 

4.  —  62«J. 

5.  I  gab. 


i^UBTRACTION. 

6.  2  7ip?/. 

7.  43(ic. 

8.  37fl;:?:l 

9.  5i«2J. 
10.  —  44.7:^^2. 


Page  32, 

11.  z^a^b. 

12.  o. 

1 3-   —lim^x. 
14.  53^^«/- 


MULTIPLICATION 


297 


15- 

$150. 

28. 

9(«— ^-f-a;). 

38. 

xy{%—ab-^c 

i6. 

25°. 

29. 

5(«  +  J). 

-d). 

17. 

$420. 

30. 

-7ix'-yy 

39- 

c(2a  +  bm  +  d) 

18. 

4xy—6a. 

31. 

$600,  A's. 

19. 

iS^+i6am. 

32. 

60°. 

20. 

i8a;2  +  ?/2-f6«. 

rage  34, 

21. 

13^^  +  ^— iC 

I- 

■3.  Given. 

—5^  +  3^^- 

JP«rflre  33, 

4- 

b—c  +  d—m. 

22. 

gcd—ab—2m 

33. 

Given. 

5. 

Sx-\-y—ab 

+  3^  +  4^. 

34. 

{2h  —  C-i-d)3^. 

+  4^. 

23. 

iSm— 23. 

35- 

(ab—c—d 

6. 

2a—b—c-{-x 

24. 

12a;' — 13a;. 

+x)y' 

+y+d. 

25. 

i6«Z'+  i^c+d. 

36^ 

a^d-b  +  c). 

7. 

a—b-{-c—a 

26. 

a—b  +  c. 

37. 

x(ab—2iC—d 

•\-c-^c—a-\-b. 

27. 

6(a  +  b). 

-m). 

MULTIPLICATION. 


rage  36, 

1-4.  Given. 

5.  42abc. 

6.  ^^abcxy. 

7.  2>dmxy. 

8.  dibcdxyz. 

9.  ^6abxy. 

10.  42acdx. 

11.  $4bcdm. 

12.  d^adfxyz. 

Page  37. 

13.  Given. 

14.  —  45rt5a:^. 

15.  42abcd. 

16.  i^2abcxy. 

17.  —  41 4abcxy. 

18.  g4^bcdxy. 


Page  38. 

19-21.  Given. 

22.  i$x^y^. 

23.  24aW. 

24.  a^x^y^. 

25.  «2j'«+«. 

26.  dx^yH. 

27.  i8«5^V. 

28.  18. 

29.  240. 

30.  6a^^. 

31.  — i8a'*J%. 

32.  4^^1/2^ 

33.  2iaW. 

34.  4oc^xy. 

35.  —  28a4^3. 


36.  —6x^y^. 

37.  2jaWc^, 

38.  —  28a3c*. 

39.  a:2^V. 

I,  2.  Given. 

3.  6fljca;2  +  8c2d 

4.  i^aWx—6acdx 

+  3«^- 

5.  —8^25^ 
-\-6ab^d—2bdm> 

6.  — 15«% 

+  2oa2^cH-ioaV 

7.  8.  Given. 

I.  6aa;  +  3Ja; 
+  2ay  +  Jy. 


298 

DIVISION-. 

Page  40, 

24.   24a^x^—6a^yK 

16.  a^b^+2adcd 

2.  2,a^+4ay 

25.    6/2_|_^^^_j_^^^ 

+  c^d^. 

—lbx—\ly. 

+  ^ca;2. 

17.  9a^—4y^* 

3.   \2M—2>cd 

26.  Given. 

18.  rz;4__^2. 

—4ad  +  ac, 

27.    ^2_^2. 

19.  x^—2xy'^-\-y^. 

4.  6ixt/—2ab 

28.     «^  +  a3_|.^_|.j^ 

20.  4«^— ic2. 

'\-6cxy—2ac, 

29.  x^-\-^x^y 

5.  z^x-\-4'bx—cx 

+3^y'+f' 

Pwgre  45. 

—Z(iy—Aby-\-cy. 

30.  a2'^  +  2a'^Z»« 

6.  ^ax-^^ay  +  az 

4-52-. 

I,  2.  Given. 

•\-ibx-\-zhy-\-lz. 

31.  a;2  4-2:ry  +  2a;2? 

3.  36. 

7.  i^cdmx—Galm 

+  ^2+2y;2;  +  ^2. 

4.  24  chickens. 

—  2icdnx-{-gah7i. 

5.  Given. 

8.  24abcx-\-i2mx 

6.  48. 

—Z2ahcy~\(imy, 

JPagre  45. 

7.   i960. 

II.  labd^xyz"^. 

I.  a^-{-2a+i. 

8.  I960. 

12.   \iaUH'''^\ 

2.  4a^  +  4a+i, 

9-  25. 

13.  aV+^ 

3.  4«2— 4c^^  +  ^l 

10.    i6w~ 

14.  c:z;(«4-^)^. 

4.  2:2+2a;y  +  ?/2. 

II.  56. 

15.    5^(«_^)5. 

5.  x^—2xy\-y\ 

12.  77. 

16.  abc(x-{-yY'^''. 

6.   I— ;rl 

13.  36  apples. 

1^,   ~3^(«  +  ^)^. 

7.  49^^-14^^+^^ 

14.  70  sheep ; 

8.   i6m^—gn^. 

100,  both. 

9.  ic4_^2. 

15.  16  and  12 

P«9re  41, 

10.    I — 4gx^. 

16.   24  plums. 

20,  a^-{-h^. 

II.   i6:z;2— 80:4-1. 

17.  42. 

21.  a^^^s^H^**. 

12.   25^2^105+1. 

18.   144. 

22.  a:^  +  a;2-j-i. 

13.    I  — 2a; +  2:1 

19-  2if;  i4f 

23-  3^  +  4^^i/ 

14.    i+4^  +  4''i'^- 

20.   i4yV  bu.,  one; 

—  I3z2— 4:z;2^2 

15.   64b^—48ab 

6^  bu.,  other, 

+  222:^-30. 

+  9«^. 
DIVISION. 

Page  47. 

4.   5^^. 

7.  sab' 

I,  2.  Given. 

5.  5- 

8.  z^c. 

3.    2^5. 

6.    2/7i 

9.  4mn, 

DIVISION" 


299 


10,  II.  Given. 

12.    — 3C. 

£3.  7J. 

14.  5^- 

15.  -65. 
t6,  ydf. 
17.  —gag. 

Page  48, 

r8.  Given. 

20.  a:®, 

^i.  c^. 

22.  2:^. 

23.  4^» 

24.  — • 

y 

25.  Given. 

26.  8a^»c2. 

27.  — 6^2f. 

28.  5a5. 

29.  72;2?/. 

30.  «Jc. 

31.  2a5c. 

32.  Sx^y^z, 
S3.  Sa^c. 
34.  i2d^x^y. 

I2iC2 

a 

36.  12X^z\ 

37.  iim^n. 


35- 


Pagre  49. 

1-3.  Given. 
4.  I^+(^-\-d*. 


5-  3^+5- 

6.  35^—14-45. 

8.  —  2a;+y. 

9.  y^+z—i, 

10.  — 5a— 45  +  6. 

11.  sab—sa. 

12.  — 4a;2 — scP 
■{■ax, 

13-  a^— 5^  +  25. 

14.  i+5«— 9«^. 

15.  2a—4h—sc. 

16.  2(^  +  5)2 

+  3^(«4-2')2. 

17.  9i?r-9^. 

18.  a;(5— c) 

19.  3^2 — 2a. 

20.  a— a^+tt^. 


I,  2.  Given. 

3.  i»+y. 

4.  «— 5. 

5.  «2_2a5  +  52, 

6.  c-^-d. 

7.  a;— d 

8.  20; +  32/. 

9.  «— &. 

10.  ic+y. 

11.  a^+ah-^-l^. 

12.  3<Z  +  26. 

13.  a4-2._ 


14.  c^—2ax  +  a?. 

15.  2a.-3+4a:2  +  8a; 
+  16. 

16.  rc  +  5. 

17.  a;— 2. 

18.  c — X. 

19.  a  +  J. 

20.  2  (a — Z>). 

1.  10  yrs.,  son  ; 
46  yrs.,  father. 

2.  15,  F.'s  m.  ; 
45,  J.'s  m. 

3.  12  and  60. 

4.  12  and  45  p. 

5.  31  cts.,  ist; 
62  cts.,  2d; 
97  cts.,  3d. 


Page  52, 

6.  20  cows ; 
180  sheep. 

7.  13I,  less; 
43 J,  greater. 

8.  9. 

9.  5  hours. 

10.  8. 

11.  7. 

12.  5  of  each. 

13.  4  hours. 

14.  32>%' 

15.  10. 

16.  7  m.,  A's  No. ; 
14  m.,  B's ; 

21  m.,  C's. 


300 


FACTOEIiq^G 


17.  ^2x,  A'sm.; 
^x,  B's  m. ; 
$Sx,  C's  m. ; 
$14^,  all. 


^^'  5>  15^  and  20. 
19-   10,  A's; 
20,  B's; 

30,  C's. 


20.  8,  16,  24 

21.  24. 


FACTORING, 


T'af/e  54, 


I,  2.  Given. 


2,  3j  3j  cf^^h, 

2,  2,  $hxxxyy. 
5,  7»  aaabicc. 

3,  7.  ^?/«/;2;;a;;?. 
iixxyyyz. 

S*  5^  5^  abhcxxx, 
9'  1,  uaahccd. 
io»  5>  ^Ziifnmnnnx, 

Page  55, 

1-3.  Given. 

4.  ^  (y  +  <?  +  3^). 

5.  2a  (a;  +  ?/  -  2^;). 
2,hc{x  —  2x—'a). 
Mm  {n  —  3). 
-ja  {sm  +  2:r). 
2']cl{bx  ~  2my). 

^axy  (3.T  +  5). 

J2.  5(5  +3^^  — 4^W 

13.  x{i  -\-x  +  x^). 

14-  3  (^  +  2  -  3?/). 

15.  i9«5(a;— i). 

I,  2.  Given. 

3-  («  +  J)  («  +  J). 


6. 

7. 
8. 

9- 
10. 
II. 


4.  {x^y)(x-y). 

5.  (m  4-  27^)  (m  +  2n) 

6.  (4a  +  i)  (405  +  j). 

7-   (7  +  5)  (7  +  5). 

8.    (2«  —  3Z»)  (2«  —  3J) 

9-   (y  +  i)  (2/  +  i). 

10.  (l    —  c2)  (l  —  c2). 

11.  C:?;"*  +  ^»)  (a;»^  4- 3^«). 

12.  (2a«--  i)  (2a»—  i). 

13.  {a^ -h  h')  (a^  +  i^), 

14.  («^  +  y)  («2;8 -f- y). 


1.  Given. 

2.  («  +  re)  («  _-  x). 

3.  (3^  +  4«/)(3^-4y) 
4-   (^+2)(y-.2). 

5.  (3  +  ^O  (3  --  x). 

6.  («5  +  i)  («  -  i). 

7.  (i  +  ^)  (i  -  Q. 
(5«  +  4^)  (5«5  -  4^). 
{2X  +  y){2X^y), 
(i  +4«)(i  —  4«). 
(5  4- I)  (5 -I). 

{X^  i-  tf)  (x^  -  y2), 

(ax  +  %)  (ax  —  %). 
(«'«  4-  b")  (a^  —  5«). 


8 

9 

10, 
II. 

T2. 

13. 
14. 


MULTIPLES, 


301 


12.  (a  -f-  i)(fl'  —  0^  +  a^ 


Page  59. 

-a^ 

4.  a3  —  a^-\.  a  —  i). 

I.  Given. 

13.  Given. 

2.  (x^i)(a^  +  x+  i). 

3-  (^  -  2^)  (^  +  ^y  +  ^y 

Page  60. 

4-  a;y  +  a;^  +  y^). 

14.  (ar  +  «/)  (a:*  -  a:3y  +  a;y 

4.   (:?;— i)(a;+  i). 

-  ^if  +  ^). 

5.  (I -6^)  (I  +6^). 

15-  (a+i)(a2_-a4.  i). 

6.  Given. 

16.  {a+  i)(a«  — a8  +  a2_.^ 

7.   (b^x)(b  +  x). 

+  1). 

8.    (^  +  ;^)(^_^;2  +  6Z22-;23). 

17.  (I  +^)(i-y  +  ^^)- 

9.  («  4-  b)  (a^  -  a*b  +  a^^ 

18.  (i  +a)(i  _«  +  a2_«? 

--aW  +  «54  _  j5). 

+  A 

lO.    (.T  +  i)  {t^  —  X^  -\-X—  l). 

19.    (i  +  J)(i_J  +  J2_63 

II.  (i  +  a)\i  ^a-\-a^  —  a^ 

4.54_^5_|_  J6). 

4-  a*  -  a^). 

20-28.  Given. 

DIVISORS. 

Page  61. 

Page  63. 

7.  a  +  J. 

I,  2.  Given. 

I,  2.  Given. 

8.  J  4-  2. 

3.  ^' 

3.  3«^- 

9.  iP  +  3- 

4.  J. 

4.   2«a:y. 

10.  a  —  2. 

5.  «<7. 

5.   4a^T^z\ 

II.  «  +  3. 

6.   20;. 

6.  6aa:2;2;2. 

12.  a;  4-  I. 

7.  7^i- 

13.  a  —  h. 

8.  6al). 

Paflre  «(>. 

14.  a  —  5. 

1-5.  Given. 

15.    a;3  4.  3a;2  4.  32: 

6.  a;  — 2/. 

+  1. 

MULTIPLES. 

Page  68. 

7.  si5x^f2^. 

13.  Q^  +  a?'— a;— I. 

I,  2.  Given. 

8.  Z^mhiY* 

14.  6a^  4-  iia* 

3.  56fl4^,2cs^. 

Page  69. 

-3«-2. 

4.  2>oxY^' 

5.  9oa3^%'5. 

6.  42oa^J*. 

9-11.  G 
12.  a*  -f 

_  7,; 

iven. 

—  2. 

302 


BEDUCTIOK     OF     FBACTION^S. 


REDUCTION 
Page  74, 


1-3.  Given. 
I 

d' 


I 

a  —  h 
x  —  y 


zy  —  3^, 

2X  —  2Z 

I 
X 

I 


OF    FRACTIONS. 


X' 

'-\-f 

X 

a 

+  x 

I 

a 

—  I 

I 

x  +  y 


3-  h 


Page  75. 

1.  Given. 

2.  a  —  x. 

a 
4.  h  ^  c. 

5.5  +  .  +  ^--. 

6.  a  —  h. 

7.  5  +  7 

b  —  a 

8.  a  +  a;  + 

9.  3a;  +  I  - 


« 


fl^  —  X 


I.  2.  Given. 
^    A^y  —  h 


,     X^  —  X 
o. • 

a;  4-  I 

i2«c  —  a  +  5 

'• — 3^ — 

5^ 


J'asre  76. 


I.  Given. 
i2ywa; 


2. 


6m 

24a%x 

iSac^  4-  24b(^ 

6^ 
a^  —  y^ 

66?V?/  —  45a;2y 
3«2  —  2b 

Page  77* 
I,  2.  Given. 
ab 

2I«2 


49« 


^  —  y^ 


^'  Q?  —  2a;?/  +  y^ 


6. 


32Q;3(a:4-y) 


REDUCTION     or     FRACTIONS. 


303 


Page  78, 
I,  2.  Given. 

3 


2CX       2ld       CpX 

2dx'  2dx^  2dx 


6. 


lO. 


II. 


12, 


ac^y  2hxy  2(^x^ 
zc^xy'  2(?xy^  2c^xy 
2c?  4-  2db         2>^x 

x^  —  2rry  +  y'^ 

a::^  -[-  2xy  +  ^2 

«2  -{-  «5     15^  —  3 
3«      '       3«^      * 

2Wc—2T)^d     :^ac—T,ad 

2it^c^l6^d 
zWc—^d 
2'bxy     hz     4az 

2bz  '  2bz'  2bz 

ax  +  ay        6x  -\-  6y 


2X  +  2y  2X  +  2y 


2X^ 


^y 


2X  +  2y 
c?  —  2ax  4-  a^ 

a^  +  2ax  -\-  a^ 
a^  —  Q? 


2. 


Page  79. 

I.  Given. 

2acx    4i^c^     hxy 
4dcx'  4bcx'  4bcx 

2,(?d     2'bcx    2>bxy 
^ay    4by    zcy 


5- 


\2X 

12?/'    12?/'    12?/'    \2y 

4abc^  Sod    bx^y 
i6a^     jSa^c      24a; 


6.  —- ^  ,  — 


24a^c'  24a^c'  240^0' 

24a^c 

ac     2cd    2xy 

2bc'  2bc'  2bc 


8. 


10. 


II. 


13. 


(a-^by    (a-bf    a^^W- 
a^  —  b^'  a^  —  W'  a2_j2 

4xy{x^y)     6a{x-hy) 
6xy(x-\-y)'  6xy{x  +  y)' 

abxy 
6xy{x  +  y)' 
ad      bx 
aW'  aW 
b'^cdx     acdm 


aWchV  aWcH'  aWM 
'^^     ayz  +  byz     dy^ 


x^z 
xy\ 


xyH 


'  xyH 


4cmx^  -\-  4cnx^ 

6acm  —  6acn    :^a^m^x 
i2a^cx^      '  i2a^cx^ 


304 


ADDITION     OF     FRACTIONS. 


ADDITION 


Page  80. 

I,  2.  Given. 


2xy 

Sgdxz 

Sabc 

X 

ya  +  2b 


7.  Given. 

rage  81, 

8.  Given. 


10. 


II. 


12. 


13. 


4$a  +  12:?:  +  20^ 

60 

Sax  -]-  6  -\-  <)ay 

12a 

ah  —  ac  +  hx  -\-  ex 

z^  —  y 

2xy 
2a  +  2ax  4-  3 
ay 


OF    FRACTIONS. 
am  —  dy 


ax  —  ay  +  ahx  +  «5^ 

5g<^  +  3Q^^  +  zbda? 
^'  iSdx 

^ah  —  2dn  —  d^ 


16. 


17. 

18. 


my 
nx  —  mx 


my  —  7iy 
19.   —  6. 

4adx  +  6hcx  —  Mm 


20. 


I.  Given. 


bdx 


2.  a  +  c  + 


hx  -f  2d 

2X 


3'  x  + 


am  —  ay  —  dx  -\-bd 


bm  —  by 


4.  $d  +a  -\-  b  —  c 


xy  +  z 


Sdh 


5-  5^  + 


2flj  —  by 
2b~^' 


Page  82. 

6.  Given. 

^bd  4-  2« 

7.  ^-^— . 

8  ^  +  ^  —  4gy. 

c 

X  -\-  y  ^  a^ 

a 

3^2  _  2a;y  —  y^  4-  a  —  b 

x-^y 

X  —  y  —  a^  -i-  Sab  —  5^ 

II.  ^ 7 "^ — 

a  —  b 

2x^-\-2xy— 2x—2y4-a-\-h 

X—  I 


10 


12. 


MULTIPLICATIOK     Of      FRACTIOi^S. 


305 


SUBTRACTION    OF    FRACTIONS. 

rage  84. 


rage  83 

I,  2.  Given. 

gahc 


3- 


4- 


d 


5,  6.  Given. 

ay  —  dm  ■\-  hm 

^'  "         my 


8. 


hy  —  f/y  -f-  J?;i 
12 


/i?/  4-  7im  4-  dm 

10.  -^— ^= ~ 

my 

h  —  my 
II. ^. 

y 

,  M  +  c7t 
12.  a  A 9 — • 

cd 

6«  +  55  —  2^/ 


13- 


14. 


«<?  + 


5<?  + 


ex 


15.  flf 


bd—dx-^ly  —  xy 
2X  +  3<Z«/ 


2V 


,    a^  —  y^  —  loa  +  lob 

16.  -^^ -^ 

10:?;  -\-  loy 


MULTIPLICATION    OF    FRACTIONS. 


rage  85. 
1-4.  Given. 

5.  h  +  ^d. 

.    ab 

6.  —  • 

4 

6ca;  —  9c?/  +  4dx  —  6dy 


ISC  +  4d 


8.  2abc, 
a  +  b 


10. 


4  +  5^ 

2«2  _f-  2^25 


Z>  +  I 

11.  8a;2  +  122;. 

12.  2ax  —  2bx  +  3«  —  35. 


13.  abc, 
a  +  b 


5 

5 

9c  — 3^ 

• 

4 
17.  3^y  +  3^- 

18      ^' 


rz;  — j5 


Page  87. 

I,  2.  Given. 
3.  6xy. 


306 


MULTIPLICATIOK     OF     FRACTIOKS 


4. 

5- 
6. 

7. 

8. 

9> 

II. 

12. 

13- 

14. 

15- 

16. 

I. 

2. 

3- 

4. 

5- 
6. 


'ay 

0?  —  y^ 
yh  +  «/;2;2* 

3 
8«  -1-  24a; 

^  (o?  +  ^) 

10.  Given. 

y  — 3^ 

2xy 
h  —  2a 

2,ab 

ly 
xy -\-  2X -\-  y'^ -\-  2y 

xy 
xS  —  y^ 
~^ 
2&  +  4. 

Page  88. 

Given. 

abdx 

dbd  4-  acd 
xy 

mx  +  nx 

— — —  • 

4 
4«c  +  4cA 

sr—  I 


8.   7a;  —  7«:r. 
aco;  —  acy 


10. 


12. 


13 


14 


Zac 


2aa^  4-  2«:p 

3^  —  3 
Sx^y 


a-\-b 
6am 


X  ■{-  I 
2dbxy  4-  y^xy 


4«  4-  6 
15.  I  —  w. 

^cx  —  3^a? 

1,  • 

4 

2.  3a;(?/4-  i). 

xy  -\-  2X  -\-  y^  -\-  2^ 
^'  ■"  ^^^ 

4.  9a:. 

S-6- 

6.  X, 

x  —  z 

8.  6ay. 

9.  a?-5^3 

2i?7  4-  y 
10.    -^- 

5 

X/^  —  IP 

12.  254-4. 

13.  2a  (c  4-  d). 

y  —  3^ 
2xy 

15.  y"' 


14 


DlVISiei^     OF     FRACTIONS. 


307 


I 

5- 
6. 

7- 
8. 

9- 
lo. 

II. 

12. 


-4.  Given. 

2« 

Wo' 

a; 

Vb 


DIVISION    OF    FRACTION 


4.  Given. 


7.^. 
21/ 


8. 


a;  —  I 
a^  —  a 


10. 


II. 


12. 


h  -\-c 

Given. 
3  times. 
5- 

9.  Given, 
a^  +  20?  +  I 
a?'  —  20^+1 

oc^  —  y'^ 

^  —  xy^  +  ^y  —  ?/3 


2x^y 

3 

11.  6. 

12.  - 


2a  +  2^ 

^     z^j^zy^ 

abcxy 
X  —  a 

15. 


3<2:2;  +  3^^ 


a  —  h 


13- 


16.  -4^ 


3«2J  +  ilz 

18.     — . 


'   4^y° 

I.  Given. 
abdmy 

ex 
mx  -f  wa; 

ax  +  a;2 

253^2  —  ^xy 

y 

6    50^^  +  5« 

rr  +  I 
7.   32;  — 3fl5:r. 


jPasre  9fl. 


5^ 

243^J/^' 
24^^ 

x-\-y 

22,XZ 


308  SIMPLE     EQUATIONS. 

U  (x  —  y) 


6. 

c 

7.  Given. 
a 


8. 


£0. 


«  4-  a; 
c  +  2 
3«  (c2  __  ^) ' 


2C 


11 


a2  _  «c  4-  c2 
12    4  (q^^  —  zaa;  4  a^) 
IX 

13- 


(«^  4  hf 

X 

a^  -\-  2X  •\-  I 
c 

X 


SIMPLE    EQUATIONS, 


Page  97. 

5.    24f 

12.   24. 

I,  2.  Given. 

6.  Given. 

13.    14. 

3.  a  —  J  4  c  —  ^. 

4ad 

14.   -i^. 

4.  a  4  ^  —ab  +  c. 

15.  Given. 

8.   '^^- 

2h  —  c 

] 

t6 

Page  98, 

9'   7i* 

5.  Given. 

i6c 

Page  102, 

6.  2  —  a  4  5. 

10. 

15 

16.    12. 

7.  ^>-|-c  — «  — 3. 

17.   60. 

8.  «6Z  —  ^c  4  2m 

18.   4. 

—  8. 

Page  . 

101. 

19.   20. 

9.   17  —  sai  —  d. 

I.   8. 

20.  aJ  4  ac. 

10.  4^6?  4  6?  —  3M 

2.   50. 

dn 
21.  — • 

—  I. 

3.  30- 

a 

II.  32  —  c  4  6?. 

4.   9. 

6c 

12.  II. 

5-  7. 

22.     T' 

3^4  2b 

6.9. 

ad  — be 
23. • 

7.   72. 

^        ac 

Pagre  100. 

8.  3a 

2ab 

I,  2.  Given. 

9.  5- 

24. 

ac—  2C 

3.   15. 

10.   28. 

5«  4  13-^ 

4.   12. 

II*    12. 

^*          24 

ONE     UKKKOWK     QUANTITY. 


309 


24^  +  I. 

50c  —  2005 

52 

b^ 

26. 

0  ^r 

3«  —  6 

45 

a— I 

2 

27. 

4 

^'-       2 

28. 

I 

32.  7*- 

2a  —  I 

a^(c  — a  ■}- en- 

-ac) 

3Z'  34. 

29. 

C2 

rage  103, 

Page  105. 

Page  106. 

34. 

20. 

I.  Given. 

15. 

13- 

35. 

24. 

2.  $8,  vest ; 

16. 

30.  days. 

S^' 

I. 

$32,  coat. 

17. 

240  m.,  one; 

37- 

If 

3.  $1500,  A; 

180  m.,  other. 

38. 

i6f 

$3000,  B ; 
$4500,  C. 

18. 

12  in.,  one; 

39- 

lA- 

4.  40  men; 

16  in.,  other. 

40. 

36. 

80  boys; 

19. 

$25,  H. ; 

41. 

II. 

880  women. 

$175,  0. 

42. 

1200. 

5.  40  miles ; 

20. 

8h.  24m.  A.  M. 

43- 

i4f 

80  miles. 

21. 

9 A  days. 

44. 

5- 

6.   1 33 J  barrels. 

22. 

32  of  each. 

45- 

4f. 

7.   12  p.,  ist; 

1 

23- 

30,  75,  and  45. 

46. 

!(.- 

^). 

24  p.,  2d ; 
60  p.,  3d. 

24. 

25  cts.,  ch.  ; 

47. 

2a  —  25 

4-c 

8.   28f  feet. 

75  cts.,  goose; 

26 

9.  $120. 

$1.50,  turkey. 

48. 

3a  — 6 

10.  $50,  B's  sh. ; 

25. 

8  ft.  8  in. 

4 

$100,  A's  sli.; 

26. 

40  and  60. 

66? 

$150,  C's  sh. 

ad      , 

49. 

5^* 

II.  18  yrs.,  w. ; 

27. 

— . — -7?  less. 
c  +  d' 

50. 

I  —  8« 
I  +  8a 

7,6  yrs.,  m. 
12.  $3000. 

ac              , 
c  +  d'  ^''''^'■ 

a 

13.   16  and  41. 

28. 

$1128. 

^i- 

2. 

4 

14.  $6000, 

310 


SIMPLE     EQUATIONS, 


Page  107. 

29.  Given. 

30.  60  lbs.,  b.; 

1 20  lbs.,  m. 

31.  isyrs.,  B's; 

30    ''    A's. 

32.  32|yrs.,  C's; 
37l    "     B's; 

40J    "     A's. 

33.  1 1 75  votes  d.; 
1325     "     s. 

34.  164  artillery; 
472  cavalry; 
564  infantry. 

35-  ^533i  B's; 

$633^,  A's; 

Ussh  O's. 
36.  $56.25,  one; 

^93-75>  other. 

rage  108. 

37-  h3^>  V^-  one. 

I280,  "    other 
SS,  i4yrs.,y'ngest; 

16   "    next; 

18  "    eldest. 

39.  i6|days. 

40.  550. 

41.  30  and  18. 

42. 

13 
43.  225  acres,  A; 

315     "     B. 


44.  2f  hrs. 

45.  5,  istpart; 
8,  2d     " 

2,  3d     " 
24,  4th  '* 

46.  9. 

47.  47  sheep. 

48.  $120. 

rage  109, 

49.  60  min. 

d 

50. 

m  —  n 

51.  300  leaps. 


52. 


a  —  c 
he 


one. 


,  other. 


53.  72  lbs. 

54.  36  hours; 
312  miles. 

55.  2oyrs.,  s.; 
40  yrs.,  f. 

56.  280. 

57.  $324,  ist; 
I108,  2d ; 
$144,  3d. 

58.  Given. 

Page  110, 

59.  8,  ist  part; 
12,  2d     '' 
16,  3d     ** 


60.  9  in.  and  1 2  in 

61.  $75. 

62.  27  days. 

63-  ^J575>  one; 
$2625,  other. 

64.  12  days. 

65.  $720. 

dd,   $384,  sum ; 
I162,  A'ssh.; 
$118,  B's  *' 
$104,  C's  " 

67.  Given. 


Pasre  m. 

(i%,   6  and  8. 
69-  3456,  one; 
2304,  other. 

70.  3  m.  an  hour, 

71.  400  in., 
or  33i  ft. 

72.  8  k.  of  one  name 
6  k.  of  another; 
3  k.  «   " 

2  k.  "   " 

73.  7  and  8. 

74.  240  leaps  of  d. 

75.  100  days; 
30000  m.,  ist; 
24000  m.,  2d ; 


SIMPLE     EQUATIOifS. 


311 


Page  114:. 

1.  Given. 

2.  x^%  y=4. 

3.  ir=i2,  ^=6. 

4.  a;=i8,  «/=2. 

5.  ic=i,  2^=3. 

6.  rr=i6,  «/=35- 

7.  tr=3,  ?/=2. 

8.  Given. 

9.  .T=4,  2/=5- 

10.  x—df  y=i2. 

11.  a;=5,  y=6. 

12.  ic=io,  ?/=3. 

13.  a:=ii,  2^=9. 

14.  a;=3,  2^=2. 

Page  116, 

15.  16.  Given. 

17-  ^-3.    2/=5- 

18.  a:=:4,     y=T, 

19.  a;=7,     «/=2. 

20.  x=i6,  2/=35- 

21.  a:=3,    ^=2. 

1.  a:=4,  «/=5. 

2.  ir=8,  y=2, 

3.  a;=5,  ?/=3. 
4-  ^=Zy  y=4' 

5.  ir=3,     ?/=4. 

6.  0^=12,  y=3. 

7.  ^=3.    y=5' 

8.  t?;=4j    5^=3- 


TWO    UNKNOWN    QUANTITIES. 
Page  117. 

9-  x=S4>  y=4^' 


10.  a;=4,    ^=2. 

11.  x=i6,  ^=7. 

12.  a;=8,    y=i» 

13.  a;=6o,  2/=36. 

14.  x=io,  y=z20. 

15-  ^=5?    y=2. 

16.  x=2,    y=4. 

17.  a.=8,     y=6, 

18.  a;=4,    ^=9. 

19.  ic  =  6, 

«/=I2. 

20.  ir=i8,  y=i4. 

1.  a;=43,  ^=27. 

2.  4  cts.,  lemons ; 
6  cts.,  oranges. 

3-  233  v.;  142  v. 

4.  21  and  54. 

5.  $48,  cow; 
$96,  horse. 

6.  40  1.;  50  g. 


7.  3  and  2. 

8.  mil  m.,  one; 
9999  m.,  other. 

9.  56. 

10.  $320,  B's; 
$250,  A's. 


12.  I900,  A's; 
$2400,  B's. 

13.  31  and  17. 

14.  $6000  h. ; 
$2500  g. 

1 5.  30  and  20. 

16.  $560,  B's ; 
$720,  A's. 

17.  25  y.  and  35  y. 

18.  I180,  ist; 
$115,  2d. 

Page  119. 

19.  140  m.,  ship; 

1 60  m.,  steamer 

20.  12  and  18. 

21.  108  ft. 

22.  3oyrs.; 
13  verses. 

23.  10  1. ;  30  g. 

24.  3  oxen ; 
21  colts. 

25.  53- 

26.  $5000,  B's  cap.; 

$4800,  A's  " 

27.  $21  or  63  g. 
an 


28.  x  = 


y  = 


n+i* 

a 
n-\-i 


3VZ 


GEKEBALIZATIOK. 


THREE   OR   MORE    UNKNOWN    QUANTITIES, 


rage  121, 

6.  ir=24,  y 

=  6,     ^=23. 

I.  Given. 

7.  ^=7i 

y 

=  10,  z=^. 

2.  x=7,    y=s, 

Z=4* 

8.  ic=24,  y 

=  60,  ;z=i20. 

3.  x=2,    y=s, 

2=5- 

9-1 1.  Given. 

4.  x=S,    y=4, 

Z  =  2. 

12.   W=2 

,  X-. 

=3»  y=A,  ^=5 

5.  x=4,    «/=3, 

z=S' 

13.     X=2 

>  y-- 

=3,   ^=4. 

4.  630  men,  ist; 

7. 

18  =  ist; 

rage  123. 

675    "     2d; 

22  =  2d; 

I.  12  yrs.,  ist; 

600    «     3d. 

io==3d; 

15    "     2d; 
17    "     3d. 

2.  I5,  s. ;  $4, 1- ; 
$20,  c. 

5.  50  cts.,  ist; 
60    "     2d; 
80    "     3d. 

8. 

40  =  4th. 
46  m.,  A's; 
9   "    B's; 

7   ''    C's. 

6.  105  min..  A; 

9- 

$64,  A's; 

3.  5,  8,  and  11. 

210    "      B; 

I72.  B's; 

420    "      C. 

I84.  C's. 

G 

ENERALIZATIO 

N. 

rage  125. 

12.  22|-hrs. 

24. 

I5625. 

I.  3  chickens. 

13.  I67.32. 

25. 

1842I  A. 

2.  30  rods. 

Page  129. 

26. 

$55.80. 

3.   12. 

27. 

$190,32. 

4-  S^TTrFS- 

14.  9077. 

28. 

I1253. 

5.  7  ft 

6.  34. 

15.  1x036.12^. 

29. 

$418.60. 

16.  587.19  bu. 

30- 

$5250. 

7.   I205,  $187. 

17.  i2f  per  cent. 

1 8.  40  per  cent. 

31- 

32. 

$3865.86. 
$1339.29. 

Pagres  127,  128. 

8.  $961,  A's; 
$614,  B's. 

19.  60  per  cent. 

20.  $3000. 

Pages  130-133. 

33- 
34. 

35. 
36. 

$222.22+. 

2i  yrs. 
Given. 
3h.  i6T^m.p.M. 

9.   1248,  g.; 

21.  137500- 

37- 

6h.32y8ym.  P.M. 

902,  1. 
10.  4f  days. 

22.  I31250. 

23.  $2700,  B's; 

38. 

9h.49-^m.P.M. 

II.  5|lirs. 

$230C 

.  C'8. 

INV0LUTI02«" 


313 


Page  137. 

2.  aW(^' 

3.  aWc^, 

4.  .T^^/V. 

5.  a^b^c^, 

6.  i6x^y*. 

7.  2\6a^h^. 

8.  62sa>W(^. 

9.  64«i2^,6^i2. 
10.  aW(^(P. 


II.  a;y5f. 


INVOLUTION. 
rage  138 

12.  (a+^y^. 

13.  (« +  ^')^ 

14.  (a:  — 2^)'««. 

15.  (^  +  ^r- 

16.  («3  4-J3)2. 

17.  aWii2. 
2'ja^^ 


19 


20. 


8^3 


21.  - 


22. 


23 


49^ 

2»» 

rtvnn'lMy.n 


26.  ic3_j_6/p2^  _j_  62^3 
4-120:^2^242:^ 
+  i2X-{-2>y^ 

+  243/2 +  24^ 
+  8. 


1.  «4  +  4fl3J  +  6aW  +  4a5s  +  J*. 

2.  i«5  _  5^4§  4.   loaW  —  IOa2^,3  _{.  50,^4  _  J5. 

3.  c^  +  1(^d  +  2ic56?2  +  356^^  +  z$(^d*^\-2lc^d^-{■^cd^+d^, 

4.  ic^  -f  6a:5y  +  i5.T^?/2  ^  20:2:^2/^  4-  isa:^?/^  +  6xy^  4-  ?/^. 

5.  x"^  —  7a:^y  4-  2  la^y^ — zs^y^  4-  35^^^ — 2 1 a:^^^  4-  ^xy^-  -y'^. 

6.  ^^®  4-  loyh  4-  45«/^;2;2  4-  \2oy*7?  4-  2io?/V  4-  2<^2y^^ 
4-  2\oif^  4-  120?/%^  4-  45?/22«  4.  io?/;2;3  4-  2;io. 

7.  a^—ga%  4-  36a'^^2  _  34^6^  4-  i26«5J4  _  126^4^,5 ^84^8^ 

8.  m^^  4-  iim}^n  4-  $$m^n^  4-  165WW  4-  33om'';i* 4- 46 2772^^2^ 
4-  462771^7^^  4-33om%''4-I65m3/^8  4  55^27294-11^7^*0 4- w'^. 

9.  a:'^  —  i2x^^y  4-  66a:io^2  _  22ox^y^  4-  4952:^7/*  —  7920:'^^ 
4-  924a::6/  —  'jg2X^y'^  -\-  4gsc(^y^  —  22o:x?y^  4-  66:z^yo 

—  122:7/1*  -f  7/*2. 


10.  a*4-wa"~^J-f-  w 
—  I 


I    „_o 70  ,      n—i       n 
-  fl^**  2  52  _j_  ^ ^  _ 


(?"-3  5s 


>»- 


n  —  2      7i  — 

X X 

3  4 


#4- etc. 


314 


EVOLUTION, 


13.    0^  -{.  ^x^  -\-  ^x  +  I, 

.14.   h^  —  4b^  +  6b^  —  45  +  1. 
15-    I  —  505  +  loa^  —  io«3  -I-  5^4 


16.   I  -j-  ^«  4- 


n  —  I 


a^  +  n 


71—1       n  —  2 


X 


a^  4-  etc. 


2  23 

J*a</es  143,  144. 

17.  a:34_3^2^4.3^22;4.3^^2_|_6^^2;  +  3:6-;2;2_f_^3_^2^2^_j_3^^2^^3 

18,  19.  Given. 

2o.  x^  +  2X  (y  -\-  z)  -\-  y^  +  2yz  +  z\ 
ii.  a^  —  2a\b  —  c)  -\-  If^  —  2bc  +  c\ 
22.  a^  ^  2a{x  -{-J/  -{-  z)  +  x^  -\-  2x(y  +  z)  +  y^  +  2yz  +  2;2 


23.  Given. 

ga^  +  12a  +  4 


24 


25. 


4^2  —  4ac  +  6^ 


26. 


27. 


36  —  iSSabc  +  i()6aWc^ 

49 
^(2  —  6hmxy  -f  gm^x^y^ 


m^ 


28,  2g.  Given. 


MULTIPLICATION    AND    DIVISION    OF    POWERS. 


Pages  145, 

I,  2.  Given. 

146, 

10.  a^y-h'^. 

11.  Given. 

12.  all. 

20- 
24. 

23.  Given, 
a 

3.  ««. 

4.  ars. 

13.  a;-ii. 

14.  &2. 

25. 

ay-^ 
b 

5.  b-\ 

6.  «"*+«. 

15.  C-2. 

16.  x'''y~^zK 

26. 

a 

7.  «-7^. 

8.  a^c^cR 

17.  4«&c~*. 

18.  32^2/. 

27. 

bx-"" 
a 

9.     ^6^. 

19.     I2a^C^. 

EVOLUTION. 

Pages  148, 

149. 

15.  #. 

16.  «-25. 

19. 

20. 

b'\ 

x^\ 

13.  ai 

17.    «-2. 

21. 

^.75; 

14.  tci 

j8.  a-6. 

22- 

■25.  Given. 

RADICAL     QUANTITIES 


315 


t*a(/es  151,  152, 

I,  2.  Given. 

3-  a^' 

4.  a^. 

5.  4*a;%i 

6.  20^1^. 

8.  2a'". 

9.  35a52;8, 

10.  6^2^. 

11.  2 3 a; 3^ 3, 

12.  8«t^>3, 


13-  (i3)^^V- 

14.  ya^y. 

15.  30^3^7. 

16. 1^. 

1.  Given. 

2.  ic  4-  2. 

3.  a—  I. 

4.  I  -[-X, 

5.  ^  + 1. 

6.  «  — J. 

7.  ^  +  -• 


8.  Given. 

9.  a;  +  «y  4-  2;. 

10.  a  —  2(3>  -f  I. 

11.  a^  -\-  2b  —  2. 

12.  I  —  2^2  4-  ir. 

13.  20^  —  4a  4-  2. 

14.  a 


X      y 
15. ~* 

y     » 


Page  156 

1-5.  Given. 
6.  aVb- 

7.  2a v^. 

8.  6\/xy. 

,  3  /- 

9.  6v3. 

10.  i5V^5. 

11.  ^SaV^b. 

12.  3a\^2^. 


13.  2m\/i  —  3^> 

14.  4^V^2^- 

15.  sa\^b. 

16.  6a'v/i3C. 

17.  I2a'\/ll. 

rage  157» 

I.  Given. 


RADICAL    QUANTITIES. 

4.  («3^(i296)i 

5.  v^i5625, 
v/87,  ^8. 

6.  ^^4^8^  '^T^. 

7.  v^«~^  v^i^". 

8.  ::;7^,  "^i-. 
10.  v^c^+Ty, 

II.   ^{x  —  y)\ 


2.   ^/8^. 


3.  ^/(zfl^  4-  yf- 

4.  V(«  —  2^)2. 

5.  a/9«^^- 

6.  ^8^7^. 

7.  yi6x^y^z. 

8.  l/"^. 

27 

9.  ^27  («  —  J)' 
10. 


11.  V^rti2^. 

12.  \/(«  —  bf, 

13.  a/«^. 

J><«</es  J5«,  159, 

1.  Given. 

2.  (^3)i^    (^^)i. 

3.  9^,  (i25)i 


V^l^  4-  yf. 

I,  2.  Given. 

3-  (3*)S  (4^)i 

4.  (a'»)i  (J")i 

5.  (a*)J,  (J^)i. 

6.  (ai)i,  (JtI)I. 

7.  (asr,  (^^3", 


316 


DIVISIOK     OF     RADICALS. 


ADDITION    OF    RADICALS. 


Page  160. 

1-3.  Given. 

4.  5V3. 

5-  2^/5+4^/3. 


6.  (2^  +  3a)  V^. 

7.  (a2  4-  3c)V'3«^. 

8.  (90;  4- 8a)  a/2«. 

9.  25^/2. 
10.  118V3. 


11.  30a v^. 

12.  (5^a;-|-6ic2)  ^/c^ 

13.  40!^^/?/ 


SUBTRACTION     OF    RADICALS, 


Page  161. 

1.  Given. 

2.  81/7. 


3.  4V30  — i2a/7- 

4.  52^/5-  _ 

5.  {2ix—io)^/ax 

6.  2'V^«  +  ^. 


7.  7^Z>. 

8.  95\/2&a;. 

9.  a~2. 
10.  ilVs- 


MULTIPLICATION     OF    RADICALS. 


Page  162. 

1-3.  Given. 

4.  90A/10. 

5.  aJic. 


6.   a/«^ 


^. 


7.  \acxy. 


'   8.  a\/«^. 
9.  Given. 


10.  v«^  « 

11.  42A/2. 

12.  12a. 

13.  6. 

14.  ^ax. 


15.  Given. 

16.  Vs- 

17.  2^/5- 

18.  (w  +  w)'V^//z4-yi 


19 


V" 


4.  Vs^^^  or 

«\/3«^- 

5.  3\/^. 

6.  (a^ta:)i 


DIVISION     OF    RADICALS 

7.  i2(a«/)i 

8.  35\/^. 

9.  -^V&. 


10.  2a 


V^f 


11.  («  +  5)«. 

12.  i5a:V^. 

13.  'v/a;  — y. 

14.  l6\/2. 

15.  32- 


KADICAL     EQUATIONS. 


317 


INVOLUTION     OF    RADICALS. 


Page  164, 
t,  2.  Given. 
3.  at 


4.  18a;. 

5.  Sa. 

xi     

6.  —  V2X. 

4 


7.  Saa^Vct. 

8.  9&2. 

9.  a2_j_2ay^-}-^. 


EVOLUTION     OF    RADICALS. 


rage  165, 

1.  Given. 

2.  3V^a. 

3.  2V^3:r. 

4.  ^9^«/- 

5.  'V/SR^ 

6.  \/^bc, 

'•  ^r 

8.  aci 

9.  2«^V 

10.  fl^^J^*^. 

11.  Vza. 

12.  fl«^6w^«. 


1-3.  Given. 

4.  ak 

1  a 

5.  a^c^, 

6.  («  +  h)l 

7.  Vttc. 

8.  \^x-{-y, 

9.  'V^«  +  J. 

10.    a/«  +  5  +  c. 

Pai/e  i67. 

I,  2.  Given. 

3.  a;  — 4^/9. 

4.  3- 

5.  A/7  —  v«- 

6.  31. 

7-   Vs^  +  ^3^. 


8:  a  — 5. 
9.  3'v/a  —  Vs. 
10.  4^2^  +  sVb. 

Page  168, 
1-3.  Given. 

xys/ci  -\-  Vc) 

A/3  —  I 


A/3  4- 


RADICAL    EQUATIONS 

Pages  169,  170, 

1-3.  Given. 

4.   (d  —  a—  cy. 

5-  25. 

6.  4f|. 

7.  256. 

8.  1 100.  '     I  —  fl 


9-   3^- 

10,  21. 

11.  252. 


12. 

2« 

13.  Given. 

I 
14. 


IS-  «a/}. 

16.  Given. 

17.  4. 

18.  ^. 

I 


19, 


1  —a 


318 


AFfECTED     QUADRATICS. 


PURE    QUADRATICS, 


14. 

^=  ±  2. 

2.  a?  =  ±  8. 

Page  173. 

15. 

X=:±S, 

3.  40  rods. 

I.  Given. 

16. 

X=:  ±  1. 

4.  30  rods. 

2.    X=±^. 

17* 

^=±    '    . 

5.  12,  one; 

3-  ^  =  ±  3- 
4.  ar  =  ±  4. 

18. 

a— I 
Given. 

30,  other. 
6.  $6. 

5-  a:  =±5. 

19. 

x=±s. 

7.  80. 

6.  x=  ±4, 

20. 

a;  =  ±  2«. 

8.   15,  less. 

7.  a;==±6. 

21. 

x  =  ±Vc^  +  (P 

60,  greater. 

8.  a;  =  ±  4. 

22. 

x=±\. 

9.  27  yds.; 

9.  ic  =  i  V6, 

23- 

X=i±^/w^^l^ 

I1.50,  price 

10.  x=  ±:  7. 

24. 

x=  ±26. 

10.  x=z  ±:  16. 

II.  a;  =  ±  2. 

II.  77  ft. 

12.  0;=  Jt  «• 

Page  174=, 

12.  x=  ±  16. 

13.  a;=±i. 

I. 

X^±Z6, 

AFFE 

CTED     QUADRA 

TICS. 

rages  178,  179, 

7. 

a 

Page  181. 

1-5.  Given. 

'^Tb 

I,  2.  Given. 

6.  a;  =  6  or  2. 

^/ah-{-d-{-  f,. 

3.  3  or  -  4*. 

7.  a;  =  9  or  —  I. 

V                4^^ 

4.  5  or  —  6. 

8.  a:  =  3a 

8. 

—  205 

5.  J  or  -  2. 

9.  Given. 

6.  2  or  —  J. 

7.  4  or  —  4|. 

±Vb'i-  Aa\ 

10.   15  or  —4. 

9- 

7  or  —  5. 

8.  4  or  —  I. 

II.  20  or  —  7. 

10. 

3  ±  2V—  I. 

9.  3  or  —  4-|. 

12.  Given. 

II. 

6  or  —  3. 

10.  9  or  6, 

I.  Given. 

II.  4  or  -  3f 

12. 

2  or  —  3. 

2.   10  or  —  7. 

0 

3.  2  or  —  5. 

13- 

^ 

Page  182* 

4.  3  or  i|. 

± 

2(; 

I.  3  or  I. 

5.  4  or  -  I  J. 

/                 W 
\     hd—ch-\ — z. 

2.  4  or  I. 

6.  a;  =  2. 

V                  4^ 

3-   3ori. 

AFFECTED     QUADRATICS. 


319 


4.  2  or  —  12. 

5-  liorf. 

6.  1 1  or  3. 

7.  I J  or  —  li- 

8.  -  J  or  -  iM. 

9.  4  or  2^. 


10.    i±V—a^+i' 
IT     —  m 


12.  I  or  —  if. 

13.  I  or  —  28. 

14.  10  or  —  ^. 
15-  -ior-i 

16.  4  or  —  I. 

17.  4  or  — if. 

18.  5  or  —  4f. 

19.  I J  or  —  |. 

20.  4  or  —  I. 

21.  if  or  —  f. 

22.  4ior  4. 

23.  3  or  —  if 

24.  4  or  —  i|. 

25-  |orf 

26.  f  or  —  I. 

27.  n  ±m. 

28.  3^  or  3a— 3 J. 

1-3.  Given. 


±  2  or  ±  ^2. 
±  A/3  or 

J  or  —  J. 


8.  I  or  - 

9.  4iorJ. 
10.  4  or  — 


2lf 


Pagres  184,  185. 

1.  8  or  4,  one ; 
4  or  8,  other. 

2.  $60  or  $40. 

3.  6  or  4,  one ; 

4  or  6,  other. 

4.  1 6s.,  $5  each. 

5.  5  or  —  6|. 

6.  16  scholars. 

7.  $30  or  $20; 
$20  or  $30. 

8.  60  or  40,  one ; 
40  or  60,  other. 

9.  36  rds.  length  ; 
28    "    breadth 

10.  20  in  file; 
80  in  rank. 

11.  10  lambs. 

12.  2  and  2. 

13.  4  and  I. 

14.  121  yds.  long; 
120    **    wide. 

15.  6  m.,  A's  rate; 

5  m.,  B's  rate. 

16.  120,  A; 
80,  B. 

17.  42  and  6. 

18.  4  lemons,  A; 
b      "        B. 

19.  14  ft.,  length  ; 
10  "    breadth. 


20.  12  rows; 

15  trees  in  each 

21.  52. 

22.  20  persons. 

rage  186. 

23.  8or— 10,  less; 
15  or— 12,  gr. 

24.  16  and  20. 

25.  50  and  25. 

26.  121  and  25. 

27.  12  ft.,  fore-w. ; 
15  ft.,  hind-w. 

28.  2 or  —18,  one; 
18  or  —2,  oth. 

29.  I  and  i, 

30.  3,  less; 

18,  greater. 

31.  16  or  36  yrs. 

32.  28  rods,  length; 
20    "   breadth. 

33-   15  yrs.,  A's; 

8  yrs.,  B's. 
34.  20  lbs.  pepper. 

rages  188,  189. 

1.  Given. 

2.  a;  =  4  or  3; 

2^  =  3  or  4. 

3.  a;  =  7  or  5 ; 

^  =  5  or  7. 

4.  x  =  S,  y  =z  6. 

5.  a:=io  or  — 12; 
y=i2  or  — 10. 

6.  tz;  =  10; 
y=i2^or7. 


>j3iO 


ARITHMETICAL     PROGRESSION?'. 


7.   iC  =  9; 

y  =  4or  Iff. 
9.  a;  =  5  or  4; 

^  =  4  or  5. 
ro.  a;  =  5  or  —3; 

«^  =  3  or  —5. 
ti.  a;  =  3;  2^=2. 
^2.  x=±s; 

15.  2?=  15  or  12; 
y=  12  or  15. 


3.  J. 

4.  |. 

5.  f 

6.  2fl^. 

7.  3^- 

8.  10. 

9-  i- 


Piagre  204, 

2.  4. 

3.  6400. 

4.  12. 


16.  a;  =  21  or  — 7, 
?/  =  7  or  — 21. 

17.  xz=62s; 
y=i6. 

18.  a;=  2  or  i; 
2/  =  I  or  2. 

1.  8  or  -4,  gr.; 
4  or  —8,  less. 

2.  30  yrs.,  wife  ; 
3 1    "     man. 

3.  0^  =  9^2,  gr.; 
^=±^2,  less. 

RATIO. 

10.  J. 

a 

11.  — 
2 

12.  X  —  y. 

14.  f 

2 

15.  — 
•^    3a: 


PROPORTION 

5.  32  and  24. 

6.  10  and  3. 

7.  16  and  12. 

8.  6  and  4. 

9.  48  and  of. 


\.  40  rows ; 

25  trees  in  each 
;.  40 yds, length; 

24  "    breadth. 
9  and  3. 

1 1  and  7. 
31  rds.,  length; 
19   "    width. 
±  7  and  i:  4. 

25  m.  and  23  m. 

1 2  and  4. 
3  or  —  2,  one ; 
2  or  —3,  other. 

19.  Equality. 

20.  Equality. 

21.  Gr.  inequality. 

22.  Less  inequal'y. 

23.  f  I  >  ¥• 

25.  7. 

26.  98. 


10. 

TI. 
12. 
13- 


430  r.,  length  ; 
320  r.,  breadth, 
20  r. ;  30  r. 
9  and  15. 
20  and  16. 


2.  9. 

3.  (>^' 


ARITHMETICAL 

4.  —  5- 

5.  If 

6.  .91. 

8.  43- 


PROGRESSION 


Page  207* 


9- 

15- 

10. 

44i. 

II. 

49^. 

12. 

3^'' 

a. 


GEOMETKICAL     PROGRESSION". 


321 


Page  208, 

2.  762J. 

3.  216. 

4.  1400. 

5-  25i 

6.  610. 

7.  175- 

8.  810. 

rage  209. 

1.  58. 

2.  278. 

3.  II. 

4.  —43- 

5.  2}. 

6.  -fj. 

7.  1024. 

8.  192. 

Page  210. 

1.  175- 

2.  1 130. 

3.  6. 

4.  6. 

5.  259. 


6.  13. 

7.  —II. 

8.  o. 

9-   255- 
10.  62. 

II    61. 

1.  I,  7,  13.  19.25, 
31- 

2.  3.  7  J,  12,  i6|, 
21,25^,30,34^, 
39>  43|.  48. 

1.  47. 

2.  —6. 

3.  102. 

4.  2,  1 1 1,   2I§,  31, 

4of ,  5oJ»  60. 

5.  i683f 

6.  98A. 

7.  5776- 

8.  foioo. 

J'aflre  213. 

9.  5- 


10.  6,  13I  2o|,  28, 
35i  421,  50. 
Slh  64I,  72. 

11.  12,  21.6,   31.2, 
40.8,    50.4,    60, 

69.6,  79.2,  88.8. 
98.4,  108. 

12.  975- 

14.  3,  5,  and  7. 

15.  loioo  yards,  01 
5|:  mi.,  nearly. 

Page  2 14. 

16.  156. 

17.  $62.50. 

18.  $667.95. 

19.  30c, 

20.  $1.20,  int.; 
$2.20,  ami 

2f.  20,  40,  and  60. 

22.  16.61  +  days. 

23.  30.  40,  50.  60. 

24.  3  days. 

25.  140. 

26.  $i78,]astpay't; 
I5370,  debt. 


GEOMETRICAL    PROGRESSION 


Page  216. 

1.  160. 

2.  4374. 

3.  4i. 

4.  320. 

5.  112. 

6.  —31250. 


P^ 

iges  217-221. 

3-  2. 

2. 

1 17 18. 

4.  3- 

3- 

9999. 

5.  6. 

4- 

27305- 

6.  5. 

5- 

3885. 

I.  242. 

6. 

8525. 

2.  2. 

I. 

30000. 

3-  500 

2. 

15625. 

4.  5. 

322 


BUSINESS     FORMULAS. 


5-  215- 

6.  567. 

2.  i,   2,  8,  32,  128. 

I.  4371. 

3-  7174453- 

4. 2imu. 

5-  9565938. 


6.  43046721. 


9.  $196.83,  1.  c. ; 
I295.24,  wh.  c. 

10.  $10.23. 

11.  2,  6,  18. 

12.  I4294967.295. 

13.  10,  30,  90,  270. 


Page  223, 

14.  $120,  $60,  $30 
15-  3.  15.  75.  375^ 
1875. 

16.  $108,  I144, 
$192,  I256. 

17.  1^,01  I.I. 

18.  8,  4,  2,  I. 


INFINITE    SERIES. 


Page  231. 

1.  li. 

2.  J. 

3.  I. 

4.  if 

Pagre  257. 

2.   1548.86. 

3-   I-973- 
4.   Ill 

6.  78. 

7.  .0375- 

8.  14.38. 

9.  2.723. 
10.  2906.3. 


5.  2. 

6.  9. 

7.  10. 

8.  }. 

LOGARITHMS. 

11.  .1814. 

12.  — 4.619. 

Page  239. 

14.  .0003321. 

15-  33-335- 

16.  191.77. 

18.  5.23. 

19.  1.0836. 


9-  I- 


10.     • 

a  —  I 
ir.  50  rods. 


20.  2.504. 

21.  2.124. 


Page  240* 

23-  -342  +  . 

24.  .546+. 

25.  .324-f. 

26.  Given. 


BUSINESS 
Pages  245-260 

2.  $349.60. 

4.   i6|  per  cent. 

6.  $12600. 


FORMULAS, 


8.  $840. 

IP.  $600. 

1 2.  1 6f  years. 

13.  10  years. 
^5*  2|  per  cent. 


17.  $2010.14. 
19.  $5414.28. 
21.  $2769.23, pr.w.; 


P-77: 


disc. 


22.  $6000,  pr.  w. ; 
$1800,  disc. 

24.  $1718.75- 
26.  $2125. 
28.  $2.33j. 


29.  $8.83+. 
31-  5MI^  per  ceiki. 
32.  12^  per  cent 
33'  9ts  per  cent. 
34.  0.  8  per  cents. 
^6.  $24630.54,  in.; 

$369.46,  com. 
38.  $1332. 
3?-  $6290.15. 


TEST     EXAMPLES, 


323 


40.  $905.80. 

41.  $3278.69. 

42.  $2278.48. 

44.  $6130.67. 

45.  $2767.60. 
47.  $2336.25. 
49.  $14166.67. 

51.  $14775- 
53.  $249.77. 


Pages  265-268, 


-6. 


4.  6v  —  I. 

5.  V—xy, 
7.  I-    


V? 


y 


a/2. 


10.    5 

c 
a 


2.  Impossible. 

3.  Impossible. 


I. 
2. 
3- 

4- 

5- 
6. 

7- 
8. 

9- 

10. 

II. 

12. 
13- 
14. 

15- 
16. 


Pagre  274, 

75«. 

57^+  7- 

3ff2;  +  3fl5j-f2C(^ 

7^6?  +  3Cf? 

—  io:c?/+  5rym. 


31. 

c  (3^  _  6Z»2c 

—  ccl). 

3^  (2/  —  3^ 

2x2  (2«  —  i). 

X  (a—  l). 
20,  II. 


17.  24  shots. 


EST    EXAMPLES. 
rage  275, 

18.   ±^/ab 

± 
«  +  I 


4 


19. 


20 


a  —  h 

x{zxy  +  2z), 

22.    (3^_c)(3^  — c). 
23-    • 

24.  640  rodSc 

25.  29 J  miles. 
b-i 


26. 

27. 
28. 
29. 

30- 

^  I    —   «^ 

32.  120. 

33.  $50. 


b+    I 


I. 

a  —  b 
3^2 


«^ 


34. 
35- 
36. 
37. 
38. 
39- 
40. 
41. 

42. 
43- 


44. 

45- 
46. 

47- 
48. 
49. 

50- 
51- 


Page  276. 

6  hours. 

x  =  s;  y 


2. 


85  f  miles. 

8ig.;5il. 
x  =  4',  y  =  6\ 

x=2)  y  =  ^', 

X  r=  $40  A, 

y  =  $60  B, 
z  =  $80  0. 
3  meters  f.  w. 
6       "      h.  w. 
8  ft.  one;  10  ft 


49  ana  77. 
48  meters. 
$2.46  a  meter. 


$100  horse, 
$200  carriage. 


324 


TEST     EXAMPLES. 


52.   . 

53.  65  hectares  y. 
100     "    elder. 

54.  26. 

55.  I5  1.;  $6s. 

56.  577  v.;  848  V. 

57.  9oyrs.A;45y. 
B;  yy.  C. 

58-  9  Vsj 

59.  2/  a/i  +  «. 

Page  278, 

60.  {:c^)^  (f)i. 

61.  V27  (a  —  h)\ 

62.  90  cts. 

63.  $10;  $18. 

64.  15  men. 

65.  28  persons. 

66.  I4  b.  %(k  w. 

67.  9. 

68.  «. 

69.-!-. 

^    I  — « 

70.  «  -f  J. 

71.  240  liters. 

72.  . 

73.  12  and  8. 

74.  ^180. 

75-  A- 

76.  $90. 


77.  ^321. 

78.  24  s.  $5  pr. 

79.  81. 

80.  45  m. ;  105  m. 

81.  80  A;  70  B. 

82.  Vx  —  ^7. 

83.  V3^  +  vsy- 

84.  ^2  ^  6t?  —  2^2^/ 

+  9  —  6^^+^^. 

85.  1 1 2000  Mayor; 
$1200  Clerk. 

S6.  216  g. ;  3of  1, 

l^agre  280. 

87.  8  and  6. 
^^.  8  cts. 

89.  10  days. 

90.  I1407  B; 
$469  A. 

91.  $50  cow; 
I200  h. 

92.  6  miles. 

93.  100  ft.  x6oft. 

94.  lh  12^  17, 

2  if,  26f 

95-  637!.^ 

96.  50  pair. 

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"     OF  THE 

DIVERSITY 


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Is  and 


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e  lower 

itiaustive 

ic  which 

but  that 

enabling 

e  author 

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ience  into 

products, 

lies  are  fol- 

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to  take  the 
'  Sentential 

d  through  to 
lar  and  High 
.-.  This  work 
ae  of  analyz- 
e  Btudent  to 
that  is  being 
unfolded,  to 
'.  G.  6.  Albfe, 
ool,  Oshkosh, 


York. 


AText-Book  on  English  Literature, 

Witli  copious  extracts  from  the  leading  authors,  English  and  Ameri- 
can. With  full  Instructions  as  to  the  Method  in  which  these  are 
to  be  studied.  Adapted  for  use  in  Colleges,  High  Schools, 
Academies,  etc.  By  Brainerd  Kellogg,  A.M.,  Professor  of 
the  English  Language  and  Literature  in  the  Brooklyn  Collegiate 
and  Polytechnic  Institute,  Author  of  a  "  Text-Book  on  Rhet- 
oric," and  one  of  the  Authors  of  Reed  &  Kellogg's  "  Graded 
Lessons  in  English,"  and  **  Higher  J^sons  in  English." 
Handsomely  printed.    12mo,  478  p] 

The  Book  is  divided  into  the  following  Te\ 

Period  I. — Before  the  Norman  Con 
Prom  the  Conquest  to  Chaucer's  dej 
From  Chaucer's  death  to  ElizabetT 
beth's  reign,  1558-1603.    Period 
Restoration,  1603-3660.    Period 
death,  1660-1745.    Period  VII 
Revolution,  1745-1789.     Pe: 
1789,  onwards. 


Each  Period  is  preced< 
great  historical  events  tl 
ing  the  literature  of  th 

The  author  aims  ' ' 
help  himself  to. 
their  relations  to 
the  authors'  ti 
helping-  to 
asshonlrl' 

allow^ 

8el( 

t 


.  Period  II.— 
0.  Period  III.— 
eriod  IV. — Eliza- 
th's  death  to  the 
storationto  Swift's 
th  to  the  French 
rench  Revolution, 

brief  resum^  of  the 
.  shaping  or  m  color* 


that  which  he  cannot 

places  in  the  line  and 

pil;  it  throws  light  upon 

great  influences  at  work, 

it  points  out  such  of  these 


limits  of  a  text-hook  would 

writers  of  each  Period.  Such  are 

c  traits  of  their  authors,  both  in 

of  these  extracts  have  ever  seen  the 

them  have  been  worn  threadbare  by 

iiie  pupil's  familiarity  with  them  in  the^ 

selections  are  to  be  studied,  soliciting  and 
exactmg  tii^  jm.«.i.^..u  a.h  ^y^.cj  step  of  the  way  which  leads  from  the 
author's  diction  up  through  his  style  and  thought  to  the  author  himself, 
and  in  many  other  ways  it  places  the  pupil  on  the  best  possible  footing  with 
the  authors  whose  acquaintance  it  is  his  business,  as  well  as  his  pleasure,  to 
make. 

Short  estimates  of  the  leading  authors,  made  by  the  best  English  and 
American  critics,  have  been  inserted,  most  of  them  contemporary  with  us. 

The  author  has  endeavored  to  make  a  practical,  common-sense  text- 
book :  one  that  would  so  educate  the  student  that  he  wot;dd  know  and 
enjoy  good  literature.     

'•  I  find  the  book  in  its  treatment  of  English  literature  superior  to  any  other  I 
have  examined.  Its  main  feature,  which  should  be  the  leading  one  of  all  similar 
books,  is  that  it  is  a  means  to  an  end,  simply  a  guide-book  to  the  study  of  Enj;li!?h 
literature.  Too  many  studants  in  the  past 'have  studied,  not  the  literature  of  the 
English  language,  but  some  author's  opinion  of  that  literature.  I  know  from  ex- 
perience that  your  method  of  treatment  will  prove  an  eminently  successful  one." — 
Jarms  H.  Shults,  Prin.  qf  the  West  High  School,  Cleveland^  0. 


Effingham  maynard  &  Co.,  Publishers,  New  York. 


